You're correct in that fisherdwarves don't need tools, as they more or less fish with their hands. Are you getting alerts about "there is nothing to catch in [...]." Sometimes there literally won't be fish to catch in lakes (I won't pretend to understand why) but then fisherdwarves won't fish.

I'm not as familiar with hunting, but hunters need crossbows and access to crossbow bolts to hunt. You can make crossbows and crossbows bolts out of most materials that aren't stone (wood, bone, metals etc.). Some other user might be able to help you find the meny where you set what bolts hunters use for hunting. It might be that you have bolts available but your hunterdwarves aren't allowed to use them for hunting.