The fish that fisherdwarves catch, like salmon, rainbow trout, steelhead trout, mussel, char, shad, pond turtles etc, are basically aquatic vermin that spawn in water bodies like oceans, rivers, brooks, murky pools, cavern seas and even in dwarf-made channels or artificial rivers.
As I understand it, oceans, rivers and brooks have an renewable supply of catchable fish, so if you're fishing from an actual ocean/river/brook tile then you will pretty much never run out.
Murky pools have a limited supply of catchable fish, generally turtles. If you're fishing from one of these, or if you're fishing from an artificial channel or river (which are treated by the game as the same as murky pools) then you'll be drawing from this limited pool of catchable fish, and within a few years or even seasons you should exhaust the supply. Keep in mind this happens even if you fill your artificial channels with water from a river or brook.
I'm not really sure about cavern fishing and whether it's renewable. Still, it shouldn't take too much effort to wall off a little section of it and build a few fisheries, so it might be worth doing that for a bit more food variety.
When you mention fish in the units screen, are you talking about stuff like carp, longnose gar, pike, sturgeon etc? Because those aren't vermin fish, they are normal creatures like the above-ground animals. When they die, they leave a corpse to be butchered. They can't be caught by fisherdwarves, and historically it's usually the reverse that happens. If you want to eat them, find a way to lure them into a chamber and then drain it of water. They air-drown and die, then you butcher them for meat.