While it is true that most early astronauts were aviators, specifically test pilots, it's also important to consider that it was the case then as it is now that the US Navy operates more planes and has more pilots than the US Air Force. Just percentage wise, that would edge towards more Navy pilots who use the naval terminology in their ranks (the Mercury 7 were 4 Navy pilots, 2 Air Force, and 1 Marine I think, though I could be wrong). I would assume that the culture would skew even more Naval as space flight progresses as early spaceflight was a couple of guys in a tin can to larger scale craft.

Another weird quirk too is that common military rank terms like "captain" and "lieutenant" don't line up between the Navy and the others (at least in the US). So the OG Star Trek guys would be Colonel Kirk and Captain Uhura under Air Force terminology, and that just sounds weird

I think you’d have to better define what the culture is that you think would change. For example, I’m sure some terms used on the shuttle are not used at all in other vehicles simply because of its design. I think naval aviators have generally been slightly more numerous in the astronaut corps, although only by a small number. I don’t think any ship has reached the point in size where there’s a dude who is paid to think about things and everyone else hits the buttons.

That might not even happen, though. Space isn't like an ocean where you can move around arbitrarily; craft mostly follow ballistic trajectories. As it is, it's actually more like artillery with human cargo than like aviation, let alone a boat that can go anywhere anytime.

The exceptions are craft with slow-burn engines like ion drives, which allow enough delta-V for a craft to hit more than one destination. Those still need energy, though, so they need to be near something like the sun to operate indefinitely. Over interstellar distances, a 20-year boost at millinewtons is still relatively short, and we're back to ballistic trajectories. On such a mission, if the crew is human and awake it would be more a matter of keeping everything operating as intended than deciding anything. I expect any culture that develops would be more about the off-time.

Speaking of boosts, burns and delta-V, you can see a bit of space's own culture growing already. My best guess is that the structure of a future interstellar mission would be a bit familiar to today's ISS astronauts.

I think the best reasoning for this has more to do with the practicalities of writing than with the accuracy of the speculation about future human endeavours. As you say, there haven't been any naval missions in space, which is exactly why when drawing from more familiar analogues you can find a richer vein by looking upon naval tradition instead. While fiction, and sci-fi in particular is going to involve some imagination to literally create and invent things all fiction tends to deal in with what we know and only a small dose of the fantastical to reframe it in a more interesting context.

The lack of similar real life equivalents for long missions with a lot of personnel and very large craft and opportunities for internal rivalries, promotions, ambition and rival navies with largely equivalent structures and traditions in current spaceflight, means that the work of writing about scenarios where that happens in space is going to be much harder and probably less resonant without drawing on something where all of that already exists. In addition to that, the hundreds of years of different naval traditions and rituals makes for more pomp and circumstance and delivers a ready-made atmosphere that's well understood even by the layperson as in those hundreds of years it has seeped in to the public imagination.

Tapping in to all the practical similarities between the scenarios often portrayed in SciFi and naval contexts along with all that cultural baggage makes for a much richer and more vivid atmosphere and setting within which the characters can interact with one another. This is reason enough to transpose naval tropes in to your space based science fiction story whether it makes the most sense or not for the way such endeavours might actually be organized in reality in the future.