Yep, Coast Guard has planes. Army National Guard also has planes, though not as nice as the Air National Guard.
The spacefaring navies of SF are generally called navies because their craft are more akin to naval vessels than aircraft. Living quarters and such. You don't see many aircraft with crews quarters, dispensaries, or mess halls.
If you don't like the idea of space navies, there's nothing keeping you from calling them whatever you want, but be wary of breaking convention. Potential reader backlash may ensue. The thing you don't want to do, though, is call them a thing that already has a definition.
Army is a ground force. Marines are an assault and amphibious force, and when troops are assigned to Naval vessels, they'll usually be Marines rather than army.
You can always give your space force Air Force type organization and have the ships themselves seconded to Army control. That way, a given ship would still be Navy, but you'd go through Army to deploy it, as it's essentially an integral part of that Army unit.
And, of course, politicians are likely to call anything army, navy, marines, or whatever, regardless of who they're actually talking about.
