I think most generally it's because naval analogues are probably the closest when you're talking about large space-based fighting vessels. The air force doesn't operate aircraft carriers, battleships, or destroyers. The navy, however, does (or did in the case of battleships). Those large sea based vessels often class quite nicely into a lot of sci-fi media for large ships.

The small ships you see are often based off of a carrier equivalent. Even when they're terrestrially based, it makes a lot of sense to streamline your military structure to have just one "space force", rather than trying to break it up into two entities like the "space navy" and "space air force", each with their own standards and logistical supply networks.

Because they're way more like ships than they are planes -- Planes don't stay in the air indefinitely or take long voyages, have large crews, etc -- They often treat the fighter pilot space ship people like AF though -- Like if I have a 'carrier' with a bunch of smaller ships on it

Because people don’t live in an airplane together for long periods of time. Pilots in sci fi are often aviation themed, but captains are naval because spaceships beyond our current level are closer to battleships, cruise ships, or aircraft carriers than fighter jets or passenger liners.

The purpose of Air Force is to monitor the skies, project power at a distance, and provide air superiority.

The purpose of Navy is to put a floating fortress off your shore and bombard your cities, carry around materiel, men, and aircraft, and patrol a vast volume of ocean.

So Navy structures fit the mission better, and this has been true since early SF.

They’re called “spaceships”

Because space is an ocean.

I think it often has to do with the scale of the ship. Operations on a starship with crew numbering in the dozens to hundreds might be closer to operations on a naval vessel than aviation. We don't have aircraft with that scale of a crew or aircraft that operate away from base for such a long period of time.

The other thought is, uh, well that's what Star Trek did.

A notable exception is the Stargate franchise, where Earth's spacecraft are largely run by the US Air Force.

In addition to what the others have said, in real life, international space law was based on maritime law. They even based directions on maritime law as the sailors used the stars to navigate, and that’s all you have in space to navigate with. So rockets and spacecraft call their directions the same as ships and sailing vessels, they have a port and starboard side, a bow and stern, up is zenith, down is nadir.

Fun fact the actual directions have some cool historical meanings. Nadir is the lowest point in elevation in the surrounding area, aka the bottom of the boat, and zenith is the area directly above you. So you could measure your latitude by measuring a star’s position relative to your zenith. Port was the side you docked on, because your steering oar was on your right. Starboard is a bastardization of the word stéorbord which is what the steering oar was called.

It's a ship. You use ship terms.

In the US at least, Air Force officer ranks match the Army and Marines. Sci fi tends to use those ranks for ground fighting teams.

Ship structure makes sense for a large ship. There is no similar aviation structure, the guy ultimately in charge is the aircraft is the pilot in command, generally the one flying the aircraft.

Some technical terms make more sense too. A hull is watertight, it translates well to a ship in space. A fuselage is describing the shape of the main body of an aircraft.

I'll agree with the other posters that being on a spaceship maps better to being on a ship than it does being in an airplane, but I would add to it one other maybe overlooked factor: Robert Heinlein served in the navy and clearly had a massive hard-on for it and included all sorts of pro-war and pro-military themes in his work, and that all came from the POV of the navy and included lots of specifically naval elements to what was presented, and he was a big enough presence in the field that it left a mark that lasts to the present day.

I guess the idea is that as in plying the oceans, you are looking at lengthy journeys in space requiring large vessels that can stock whatever supplies you need while protecting you from harsh elements with thick hulls. And they do talk about space fighters, which is more of an aviation term, but these are typically launched from the space equivalent of an aircraft carrier.

Aviation and marine vehicles are both flying, just in different fluids.

I wonder if it has much to do with the USAF being a relatively new service with a proportional cultural impact, coming into being as a service in 1947. Up until then, combat aviation was subordinate to the Army and Navy. This would point to a preponderance of Army/Navy WWII vets among the show's consultants and audience.

"Why do they use" is a question. "Why they use" is an explanation.

In addition to all the other reasons offered re: functional analogy, many of the aviation terms themselves come from naval / boating / sea-faring. Pilot is a good example, previously having been used in the sense of "riverboat pilot," etc.