The only possible exceptions I can think of are fish(I imagine gills and mouth are not connected but don’t really know). I am excluding bacteria and viruses and I believe they don’t really breath(correct me if I’m wrong).
Dolphins are the most classic example, having the blow hole.
Insects ‘breath’ through their ‘skin’
Dolphins can breathe thru their mouths tho.
There’s been some with damaged blowholes who have to resort to that.
What they do normally is like when you can “block” the connection of one or the other so air only goes thru one path. No idea how to describe that, or if other people can do it tho.
But if a dolphin needs to they can breathe thru their mouths just fine.
I read about this after I had the showerthought that dolphins have their nose on the back of their head. Apparently they evolved a ‘plug’ to separate the nose-lungs and the mouth-stomach. It was thought they couldn’t control it, but they found an example of a dolphin who had a damaged blowhole and figured out how to breathe from the mouth. But it looked like it wasn’t easy, iirc the dolphin had to basically swim fast before surfacing to get the air to help force it open. The dolphin wasn’t expected to live long.