They are fish. :) They still can and do get oxygen from the water, which is why you will see them breathing even underwater - this is most pronounced if they are sick, in which case they will often 'pant'. They do all that the way a normal fish does.
HOWEVER, because they originally come from hot, slow-moving water which has low oxygen content and poor oxygen exchange, this is not usually enough. As such, they developed the labyrinth organ to supplement this by oxygen from the surface. It is true that if you remove the labyrinth organ, or prevent the fish breathing from the surface, it will suffocate, but that is because it isn't designed to rely solely on one set of lungs or the other - it needs both. It will also suffocate if you prevent it absorbing oxygen from the water.
So, they aren't like dolphins, holding their breath underwater and breathing exclusively from the surface, and are in fact proper fish. :)