Yes. They are holding their breath. It's not that fantastic anyway (just a little :p). Maybe only 4 times better than a real human being. The record for holding ones breath is actually 12 or so minutes. Granted, he didn't move at all. And those pearl divers in south Asia can hold their breath for a very long time too. Having a little fight while underwater isn't that far fetched for a fantasy game, considering humans can be trained to become very good at it too.
The reason only Rikku, Wakka and Tidus can fight underwater is that the others aren't dressed for swimming :p, much less fighting while swimming. Tidus being able to swim with that sword is borderline already, Auron would sink, and so would Lulu too. Yuna isn't trained in holding her breath, as displayed in X-2 (she just set a new personal on 2 minutes or something at the beginning of the game). I think Kihmahri should have been able to fight underwater though.