Seven points to consider:
Obviously the planet Earth has a companion object, the Moon, which looks like the Moon in the sky.
Obviously at any one moment the Earth will be as far from the Moon as the Moon is from the Earth.
Since the Earth has about four times the physical diameter of the Moon, obviously the Earth appears to have about four times the angular diameter as seen from The Moon as the Moon appears to have as seen from the Earth.
So in a story set on a moon base on the near side of the Moon the Earth will appear about four times as large in the sky as the Moon appears in the sky of Earth.
And someone could write a story where the Moon has been terraformed to be habitable and have a breathable atmosphere - kept from escaping by a forcefield or something - and people could walk around outside without spacesuits and enjoy the view of Earth.
It may be noted that tidal interactions cause the Moon to very, very, very slowly increase its distance from Earth. So billions of years ago, the distance between the Earth and the Moon was just a small fraction of the present distance, and the Moon appeared many times larger in the sky of Earth than it does today, and the Earth appeared four times larger in the sky of the Moon than the Moon appeared in the sky of Earth.
Of course neither the Earth nor the moon was habitable in that era, so there were no native lifeforms to enjoy the views. But possibly advanced civilizations might discover young planet-moon systems where the two bodies are close together and terraform one or both of the worlds in such a system to make them habitable.
And of course the small planet Mars looks gigantic in the skies of its two moons, Deimos and Phobos, that orbit close to Mars. Deimos and Phobos are very tiny bodies, probably captured asteroids, so from the surface of Mars they look like very small orbs or mere dots of light.
The dwarf planet Pluto and its largest moon Charon are sort of a double planet, being so close together and so similar in size, and each appears very large as seen from the other.
Of course Deimos, Phobos, Mars, Pluto, and Charon are not habitable for any multi celled Earth life forms, though there is speculation that Mars could be terraformed to be habitable.
The four giant planets in our solar system, Jupiter, Saturn, Uranus, and Neptune, all have several times the diameter of the Earth, and all have at least one moon orbiting them at the distance between the Earth and the Moon, or closer, so obviously they all appear to be many times the apparent angular diameter of the Moon as seen from their inner moons.
In old science fiction stories the four large Moons of Jupiter: Io, Europa, Ganymede, and Callisto, were often depicted as habitable worlds, thus making it possible to encounter a Jovian goon on a Jovian moon. Saturn's largest moon, Titan (and sometimes other Saturnian moons) was often depicted as habitable. Neptune's largest moon Triton, was sometimes depicted as habitable, and sometimes even the larger moons of Uranus, even though they are much smaller.
Of course the surfaces of those moons are now known to be deadly for Earthly lifeforms, as no doubt seemed scientifically very probable even when those stories were being written.
However, scientists have given serious thought to the possibility that giant planets that orbit in the habitable zones of stars could have giant, Earth-like, moons that could be habitable.
A habitable moon of a giant planet could orbit it close enough that the giant planet would appear enormous in the sky of the moon, just as the giant planets in our solar system appear enormous in the skies of their inner moons.
And if that giant planet has two or more giant, habitable moons orbiting it, the life forms on each of the habitable moons would not only see the giant planet occupying a large part of their sky, but also see the other large moon (or moons) as sometimes a mere dot of light, and sometimes when closer possibly appearing several times as large as Earth's Moon.
There have been many other questions about possible Earth sized habitable moons of gas giant planets.
I have answered a number of those questions.
Here is a link to a previous question and my answer to that question includes a link to a question and answer which has links to previous questions and answers.
What are the day and night fluctuations for a moon orbiting a planet the size of Jupiter?1
The article "Exomoon Habitability Constrained by Illumination and Tidal heating" by Rene Heller and Roy Barnes, Astrobiology, January 2013, discusses factors affecting the habitability of exomoons.
The distances between planets in our Solar System are so vast that no planet can be seen as more than a dot of light from another planet. And it might be assumed that all other planetary systems would be similar.
But astronomers have now discovered a little more than four thousand planets orbiting other stars, some of them in systems with more than one known planet. And there are some systems where the distances between planetary orbits are many times greater than in our system, and some other systems where the distances between planetary orbits are many times smaller than in our system.
In the Kepler-70 system, the orbits of Kepler-70a and Kepler-70b are so close that:
During closest approach, Kepler-70c would appear 5 times the size of the Moon in Kepler-70b's sky.
Among potentially habitable exoplanets, exoplanets that orbit in the habitable zones of their stars, the potentially habitable planets in the TRAPPIST-1 system orbit closest to each other.
The orbits of the TRAPPIST-1 planetary system are very flat and compact. All seven of TRAPPIST-1's planets orbit much closer than Mercury orbits the Sun. Except for b, they orbit farther than the Galilean satellites do around Jupiter, but closer than most of the other moons of Jupiter. The distance between the orbits of b and c is only 1.6 times the distance between the Earth and the Moon. The planets should appear prominently in each other's skies, in some cases appearing several times larger than the Moon appears from Earth. A year on the closest planet passes in only 1.5 Earth days, while the seventh planet's year passes in only 18.8 days.
Added August 12, 2019:
I have just found a discussion of another type of planetary configuration where a large planet and a small planet could sometimes be close enough that each would appear as a disc or orb in the skies of the other planet. At Sean Raymond's PlanetPlanet site, is this post:
About so-called "horseshoe orbits". An Earth like planet in a horseshoe orbit relative to a second planet could periodically approach the second planet close enough that during the period of closest approach the other planet would appear as large as the Moon does from Earth.