They can't breathe underwater. They have to hold their breath. This is a fact, and it is stated in the game.

I realise that a character never runs out of breath while you're actually playing, but that's just for convenience - it'd be a pain if you only had a matter of minutes to explore the various underwater sites without surfacing for breath. Also, many of the underwater locations have Save Spheres throughout them. It's possible that Save Sphere replenish breath the way they replenish health.