In the case of such a planet, the day/night cycle would be equal in length to the planet's orbital period, and not to its rotational period.
What you are asking about is a planet with an axis of rotation that is in the plane that the planet orbits its star in. You might think that means the rotational axis of the planet would thus have a tilt of zero degrees, but astronomers consider a rotational axis at right angles to the orbital plane to have an axial tilt of zero degrees, and a rotational axis actually in the orbital plane to have an axial tilt of ninety degrees.
There would be one light period and one dark period in every orbit of the planet around its star. No matter how many times the planet rotated around its axis during that orbit, the star would appear to rise and set and once each planetary year.
Oops! I was wrong about that. There would be only one daylight period and one nighttime period during each year for the north and south poles of the planet.
But the farther away from the poles a place was, the more often sunlight would be blocked by the body of the planet as it rotated. So the length of days and nightss would change over the course of the year in one location, as the angle to the star changed.
It would be an interesting problem to figure out the changing lengths of days and night on various locations on the planet.
Could life develop and survive on such a planet? In many regions of the planet days and nights might last for months at a time.
The ultimate example of long days and nights is eternal days and nights. A slight difference in the mass of 2 stars will cause a larger difference in their luminosities.
So for a planet to orbit in the circumstellar habitable zone of a dim star, it will have to orbit so close that it will experience intense gravitational and tidal forces from the star, which will swiftly slow down the rotation of the planet until the rotation rate of the planet equals its orbital period around the star.
Thus one side of the planet will always face the star and have eternal day and heat, and the other side of the palnet will always face away from the star and have eternal dark and cold.
Can life evolve and survive on such a planet? That is unknown.
Astronomers for many years ruled out red dwarfs as potential abodes for life. Their small size (from 0.08 to 0.45 solar masses) means that their nuclear reactions proceed exceptionally slowly, and they emit very little light (from 3% of that produced by the Sun to as little as 0.01%). Any planet in orbit around a red dwarf would have to huddle very close to its parent star to attain Earth-like surface temperatures; from 0.3 AU (just inside the orbit of Mercury) for a star like Lacaille 8760, to as little as 0.032 AU for a star like Proxima Centauri (such a world would have a year lasting just 6.3 days). At those distances, the star's gravity would cause tidal locking. One side of the planet would eternally face the star, while the other would always face away from it. The only ways in which potential life could avoid either an inferno or a deep freeze would be if the planet had an atmosphere thick enough to transfer the star's heat from the day side to the night side, or if there was a gas giant in the habitable zone, with a habitable moon, which would be locked to the planet instead of the star, allowing a more even distribution of radiation over the planet. It was long assumed that such a thick atmosphere would prevent sunlight from reaching the surface in the first place, preventing photosynthesis.
This pessimism has been tempered by research. Studies by Robert Haberle and Manoj Joshi of NASA's Ames Research Center in California have shown that a planet's atmosphere (assuming it included greenhouse gases CO2 and H2O) need only be 100 millibars (0.10 atm), for the star's heat to be effectively carried to the night side. This is well within the levels required for photosynthesis, though water would still remain frozen on the dark side in some of their models. Martin Heath of Greenwich Community College, has shown that seawater, too, could be effectively circulated without freezing solid if the ocean basins were deep enough to allow free flow beneath the night side's ice cap. Further research—including a consideration of the amount of photosynthetically active radiation—suggested that tidally locked planets in red dwarf systems might at least be habitable for higher plants.
So it is still uncertain whether there could be life on planets tidally locked to their stars with eternal day and eternal night on different sides of the planet.
Only some spectral and luninosity classes of stars are suitable for having planets with life. So as a result, the longest possible year of a planet habitable for humans or for beings with similar environmental requirements should be only about 5 or 10 Earth years long. So no planet with an axial tilt of 90 degrees could be habitable anyway if its year was more than 5 or 10 Earth years long.
That means that the longest possible day length on a otherwise habitable planet with an axial tilt of 90 degrees would be no more than 5 or 10 Earth years long, and then only at the rotational poles of the planet. Other regions should experience shorter day/night cycles. Plenets with shorter orbital periods would experience shorter day/night cycles.
So since 5 or 10 years is a lot shorter than billions and billions of years, if life is possible on a tidally locked planet with eternal day and eternal night on different hemispheres, it should be possible on your planet with an axial tilt of 90 degrees and a day/night cycle up to 5 or 10 years (though possibly much less than that depending on the orbit of the planet).
But what if life is not possible on a tidally locked world because the day side gets too hot and the night side gets too cold? Would day/night cycles equal in length to a planet's orbital period be short enough for life to flourish?
If the question is about whether the planet would be habitable for humans, or for intelligent beings with similar environmental requirements, one place to look for answers is Habitable Planets for Man, Stephen H. Dole, 1964.
On pages 58 & 59 Dole discusses the possible rotation rates for a planet habitable for humans from the viewpoint of the length of the day/night cycle.
From the standpoint of human Habitation, there are two limits related to rotation rates. For slow rotation rates, a limit would be reached when daytime temperatures become excessively high in the low latitudes below some critical latitude and when nighttime temperatures become excessively low polewards from this same latitude, or when the light-darkness cylcle became too slow to enable planets to live though the long hot days and long cold nights. If rotation rates were increased steadily, a limitingpoint would be reached when surface gravity at the equator fell to zero and amater was lost form the planet, or when the shape of the surface became unstable and axial symmetry was lost.
Just what extremes of rotation rate are compatable with habitability is difficult to say. These extremes, however, might be estimated at, say, 96 hours (4 Earth days) per revolution at the lower end of the scale and 2 to 3 hourss per revolution at the upper end, or at angular velocities where the shape becomes unstable due ot the high rotation rate.
So Dole believed that the longest day/night cycle compatible with (human or human-like) habitability would be about 4 Earth days or 96 Earth hours long.
And it fact it is possible for a planet in the habitable zone of a star to have an orbital period less than 96 Earth hours or 4 Earth days long.
Here a a link to a list of possibly and potentially habitable exoplanets, exoplanets which orbit in the circumstellar habitable zones of their planets, and which could thus be habitable with they satisfy all the other criteria for habitabiity.
If you sort the list by length of orbital period, you will find planets with orbital periods of 12.4 Earth days or less, with the shortest orbital period, 4.05 Earth days, for TRAPPIST-1 d.
TRAPPIST-1 is a very low mass and very dim star, with an absolute magnitude of 18.4 (the higher the absolute magnitude, the dimmer the star) and a luminosity which is about 0.000553 that of the Sun. However, spectal class M9V stars have luminosities of about 0.0003 that of the Sun. So I expect that it could be possible for a planet in the habitable zone of a very dim star to have an orbital period of only 3 or maybe 2 Earth days.
But there is a problem with having habitable planets so close to their stars.
When astronomical objects orbit deep within the gravity wells of larger astronomical objects, where the gravity and tidal forces are strong, several things happpen.
Their orbits become highlely circular, their orbital planes move very close to the equatorial planes of the objects they orbit, and their axial tilt decreases to zero, thus becoming perpendicular to their orbital planes, and they become tidally locked to the objects they orbit.
So in a few million years your planet would have its axial tilt straightened to zero degrees and would become tidally locked to its star. Which is wat you don't want.
But suppose that your planet was actually a planet sized moon orbiting around a giant planet, and that giant planet orbited close to their star, a dim red dwarf star.
The giant, planet sized moon would quickly (in a few million years) have its orbital plane shifted into the equatorial plane of the giant planet, and would have its axial tilt reduced to almost zero, so that it's rotational axis would point almost totally parallael to the rotational axis of the planet, and the moon would become tidally locked to the planet.
And the moon would stay in that relationship to the planet it orbited, even if the moon and the planet were close enough to the star for planets to normaally become tidally locked.
So what are the axial tilts of the solar system planets?
Mercury 0.3 degrees, Venus 2.54 degrees, Earth 23.44, Mars 25.19, Jupiter 3.13, Saturn 26.73, and Neptune 28.32 degrees. And Uranus has an axial tilt of 82.23 degrees, almost a full 90 degrees. So Uranus almost has its axis of rotation in the plane of its orbit.
Soit wold be possible for a giant planet in another star system to have an axial tilt of about 90 degrees, and thus have its axis of rotation in or near the plane of its orbit around the star. And that would cause all of the inner moons of that planet to have their rotational axes in or near the plane of hte planets orbit, since they would be parallel to the planet's axis of ratation.
So you could get a habitable moon of a giant planet with its axis of rotation, like that of its planet, in the plane of hte planet's orbit around the star. And that would be pretty much what you asked for.
But there are some numerical factors to consider.
A giant, habitable moon of a giant plane twould have to orbit outside the habitable edge of the giant planet. If it orbited inside the habitable edge of the planet, the starlight reflected from the giant planet, plus the heat radiation of the giant planet itself, plus the tidal heating caused by tidal interactions, plus the heat from the star, would heat up the moon too much and it would suffer from a runaway greenhouse effect.
So the moon would have to orbit outside the habitable edge of the giant planet, which has been put at 5 times the radius of the planet.
In our solar system the giant planets have radii of 71,492 kilometers for Jupiter, 60,268 kilometers for Saturn, 25,559 kilometers for Uranus, and 24,764 kilometers for Neptune.
So the radii of the habitable edges of the giant planets would be: 357,460 kilometers for Jupiter, 301,340 kilometers for Saturn, 127,795 kilometers for Uranus, and 123,820 kilometers for Neptune.
So how long would it take a moon to make one orbit at those distances right at the habitable edges of those planets? The masses and the distances of the objects are necessary for such calculations.
Jupiter has 317.8 times the mass of Earth, Saturn 95.159, Uranus 14.536, and Neptune 17.147.
Here is a link to an orbital period calculator. I put the mass of the planet in Earth masses in the "mass of Sun" box, asssume that the habitable moon would have one Earth mass and put that in the "mass of planet" box and the distance to the habitable edge itothe sami-major axis box.
So according to those calculations moons orbiting at the habitable edge of Jupiter would have an orbital period of 1.37857 Earth days, of Saturn 1.94284 Earth days, of Uranus 1.33489 Earth Days, and of Neptune 1.17796 Earth Days.
Giant planets don't get much larger than Jupiter. As planets accumulate mass beyond the mass of Jupiter, their gravity condenses their interiors and makes themdensier, so their expansins slows down and stops. And perhaps planets even shrink in size as more mass is added.
The dividing line between the most massive planets and the least massive brown dwarfs is about 13 times the mass of Jupiter.
And my guess is that the radius of giant planet with exactly 13 times the mass of Jupiter would be no more than 20 percent more or less than that of Jupiter.
So a giant planet with a mass of 13 times the mass of Jupiter, or 4,131.4 times the mass of Earth, shoudl have a radius in the range of 57,193.6 to 71,492 to 85,790.4 kilometers, and thus a habitable edge in the range of 285,968 to 357,460 to 428,952 kilometers.
So a moon orbiting at the habitable edge of a giant planet with 13 times the mass o fJupiter (4,131.4 times the mass of Earth) should have an orbital period of 0.273981, or 0.382900, or 0.503336 Earth days.
Heller and Barnes "Exomoon Habitability Constrained by Illumination and tidal heating",Astrobiology, volume 13, number 1, 2013, section 2. Habitability of exomoonss, says:
The longest possible length of a satellite’s day compatible with Hill stability has been shown to be about P∗p/9, P∗p being the planet’s orbital period about the star (Kipping 2009a).
This means that a moon can't have a stable orbit around it's planet unless the orbital period of the planet around the star is at least 9 times as long as the orbital period of the moon around the planet.
As written above moons orbiting at the habitable edge of Jupiter-like planets would have an orbital period of 1.37857 Earth days, of Saturn like planets 1.94284 Earth days, of Uranus like planets 1.33489 Earth Days, and of Neptune like planets 1.17796 Earth Days.
Thus the minimum orbital period of a Jupiter like planet with a habitable moon at the habitable edge would be 12.407 Earth days, of a Saturn like planet 17.48556 Earth days, of a Uranus like planet 12.01401 Earth days, and of a Neptune like planet 10.60074 Earth days.
Since a moon orbiting at the habitable edge of giant planet with 13 times the mass of Jupiter should have an orital period in the range of 0.273981, or 0.382900, or 0.503336 Earth days, the planet that it orbits should have an orbital period in the range of at least 2.465829 to 3.4461 to 4.530024 Earth days.
And of course habitable moons of exomplanets might orbit beyond the habitable edge and thus have longer orbital periods. And of course the orbital periods of planets with habitable moons might be much more than 9 times as long the orbital periods of those habitable moons.
So some such habitable moons orbiting giant planets with axial tilts of about 90 degrees could have years years less than 4 Earth days long, and thus day/night cycles at their poles less than 4 Earth days long.
But they would be small minorities of all the habitable moons orbiting giant planets with axial tilts of about 90 degrees compared to the orbital planes of their planets. In most cases the panets would orbit much farther from their stars and so the moons would have much longer day/night cycles at their poles.
So maybe you shuldn't pay too much attention to Dole's guess that a day/night cyclelast for 4 Earth days would be the limit for a habitable world.
I note that on Earth the polar regions of periods of light during the summer lasting for weeks or months, and periods of darkness during winter lasting for weeks or months.
So I think that it would be quite possible for your world to have periods of daylight and nighttime darkess in their polar region lasting for tens of days, while their equatorial regions may have day/night cycles about an Earth day long during some seasons of their year.