I don't know if it is possible for a double planet to have 2 Earth like planets as close together as you want. Being separated by about the Earth-Moon distance seems more plausible to me, so you might need to get someone to do calculations. There would be great views of one planet from another if they were as close as you want them to be, but I don't knowhow possible that is.
You may have to worry about the Roche limits of the planets. If the planets ae within their Roche limits they will disrupt each other.
As a rule the Roche limit of the denser or more massive object is used to calculate if the less dense or less massive objects is disrupted. Your planets are twin planets, very similar. But I don't know if you can plausibly make them identical enough to avoid having one break up.
The Moon becaem tidally locked to the Earth by a process of tidal interaction which also is slowing down the Earth's rotation rate and pushing the Moon farther from Earth.
If you want your two Earth like planets to be habitable for humans, and/or have native life with similar requirements including multicelled animals and maybe native intelligent beings, the planets will have be billions of years old. And over those billions of years the processes which slowed down their rotation and tidally locked them will have also have forced them farther and farther apart.
So if they were moving farther and farther apart for billions of years, how could they end up not much farther apart than their Roche limits?
You may need to have someone do calculations.
Do the planets need strong magnetic fields to retain their atmopsheres, or could their escape velocities be enough without magnetic fields?
Titan, the moon of Saturn, has a dense atmosphere without a magnetic field of its sown, and also has a very low escape velocity.
Titan has no magnetic field, although studies in 2008 showed that Titan retains remnants of Saturn's magnetic field on the brief occasions when it passes outside Saturn's magnetosphere and is directly exposed to the solar wind.[33] This may ionize and carry away some molecules from the top of the atmosphere. Titan's internal magnetic field is negligible, and perhaps even nonexistent.[34] Its orbital distance of 20.3 Saturn radii does place it within Saturn's magnetosphere occasionally. However, the difference between Saturn's rotational period (10.7 hours) and Titan's orbital period (15.95 days) causes a relative speed of about 100 km/s between the Saturn's magnetized plasma and Titan.[34] That can actually intensify reactions causing atmospheric loss, instead of guarding the atmosphere from the solar wind.[35]
https://en.wikipedia.org/wiki/Atmosphere_of_Titan#:~:text=Titan%20has%20no%20magnetic%20field%2C%20although%20studies%20in,some%20molecules%20from%20the%20top%20of%20the%20atmosphere.
But if Titan was at Earth's distance from the Sun the solar wind would be much stronger and knock off gas particles much faster. The atmospheric gases would alos be much hotter and faster and would escape faster.
Ganymede, the largest moon of Jupiter, is the only moon in the solar system with a measured magnetic field. Ganymede is tidally locked and has a day 7.154 Earth days long. Io and Europea have shorter days and rotate faster but are smaller than Ganymede. Callisto and Titan rotate slower than Ganymede and are less massive.
And I don't kow if that is the reason why only Ganymede has detected magnetic field.
I note that if your planets ae supposed to be habitable for humans, there may be an upper limit to how long their days can be.
Sephen H. Dole, in Habitable Planets for Man, 1964, discusses what is necessary for a planet to be habitable for humans.
https://www.rand.org/content/dam/rand/pubs/commercial_books/2007/RAND_CB179-1.pdf
On pages 58 to 61 he dicusses the rotation rate for a habitable planet and guesses that between 2 to 3 Earth hours and 96 Earth hours per day would be the habitable range.
You should find out if you agree that 96 Earth hours would be the longest possible day for a planet to remain habitable for humans. You may need to increase the separation between your planets, and thus the lengths of their orbits and days, to make a plausible astronomical situation, but you shouldn't make the length of day longer than the uper limit, whatever it may be, for a habitable planet.
I note that a large moon orbiitng a giant planet that orbits in the Circumstellar Habitable zone of a star would be similar in most respects, though not all, to a double planet of twin Earth-like planets orbiting in the Circumstellar Habitable zone of a star,
And there have been scientific studies of the hypothetical habitabilty of hypothetical exomoons, as they are called.
"Exomoon Habitability Constrained by Illumination and Tidal Heating", Rene Heller and Roy Barnes, Astrobiology, Volume 13, number 1, 2013 discusses various factors affecting the potential Habitability of hypothetical exomoons.
They discuss the magnetic fields of exomoons whch may depend on their rotation rates. Most such exomons would be tidally locked to the planet and not to the star, so their day would equal their month orbiting the planet and not their year orbiting the star.
However, considering an Earth-mass exomoon
around a Jupiter-like host planet, within a few million years
at most the satellite should be tidally locked to the planet—
rather than to the star (Porter and Grundy, 2011). This configuration would not only prevent a primordial atmosphere
from evaporating on the illuminated side or freezing out on
the dark side (i.) but might also sustain its internal dynamo
(iii.). The synchronized rotation periods of putative Earthmass exomoons around giant planets could be in the same
range as the orbital periods of the Galilean moons around
Jupiter (1.7–16.7 d) and as Titan’s orbital period around
Saturn (&16 d) (NASA/JPL planetary satellite ephemerides)4
. 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). Since the satellite’s rotation period also depends on
its orbital eccentricity around the planet and since the
gravitational drag of further moons or a close host star could
pump the satellite’s eccentricity (Cassidy et al., 2009; Porter
and Grundy, 2011), exomoons might rotate even faster than
their orbital period.
https://faculty.washington.edu/rkb9/publications/hb13.pdf
It is interesting that in some cases an exomoon might rotate even faster than its orbital period. The faster a world rotates, the more likely it would be to have a strong magnetic field. So if an exomoon could avoid becoming tidally locked to either its planet or its star it would retain a rotation rate closer to its original rotation rate, and thus rotate faster than if it was tidally locked tothe planet, let alone to the star.
So you need to find out whether planets part of double planets mighht also possibly be able to rotate faster,and thus in les time, than their orbital periods around the center of gravity.
Thus it may be possible for a planet in a double planet to have moved far enough away from its partner to be consistent with the tidal interactions over the billions of years it should have taken to develop an oxygen rich atmosphere, while still rotating faster than the slowest possible rate consistent with remaining habitabel, and also fast enough to generate a strong agnetic field.
Or your story could involve a double planet which was terraformed by an advanced civilization to become habitable, and while the two planets in the pair were still yung and orbiting close together. And for the purpose of your story it might not matter how many hundeds of thosuands or millions of years ago that terraforming was, or how many more millions or billions of years the planets can remain habitable in the future.
