Other countries have naval aviation don't they? What do you think the French fly off their carriers? What do you think the British plan to fly off theirs? (The British are actually planning to fly both Royal Air Force and Royal Navy aviators off theirs.) The British Royal Marines also sometimes fly fast air, when they have aircraft, but only as part of the Naval Service.
All these countries have very different military cultures between air force and navy. Navies, people who drive ships, generally have more ridged rank structures. Airforces, groups that fly aircraft, delegate authority according to job descriptions. There are historical and practical reasons for this, mostly linked to minimum team sizes, but safe to say the two cultures can conflict. Keeping some degree of separation reduces conflicts and increases cooperation/safety.
Russia is similar. Aerospace Forces, Airborne Forces, Navy, National Guards, Border Services operate aircrafts. Army used to have own aviation, but it was recently merged under Aerospace Forces.
Don't forget the 6th one, Space Force.
In all seriousness, the various US armed forces all step over each other. The Navy has it's own air force, but the USAF also has its own navy (Google "Tyndall AFB"). The Coast guard has plenty of helicopters and the national guard has a bit of everything. They all operate different recruiters, different training schools, and maintain unique relations with foreign forces. It's a total mess compared to countries like Canada where all the forces train and fight together from day one at basic training.
When a Canadian ship sails with a helicopter, the ship is Navy. The helicopter, its aircrew, and all the servicing people, are airforce. Their navy (RCN) handles the ship stuff and their airforce (RCAF) all the flying stuff. So a US carrier could stay in the USN, while the air detachments would be handled by the USAF. This sort of cross-branch cooperation means everyone gets along. The navy doesn't get jealous of the airforce's new toys because everyone plays together. When favors are needed, everyone has buddies in the different branches and knows each other's abilities.
This is a uniquely American issue. The US has four or five different air forces. They all fight amongst each other for research and new toys. Other countries operate on the principal that "if it flies, it is airforce", which moots most all of this debate,