I haven't tried 1-level artificial pools, but as far as I can tell, fishing out one pond does indeed fish out the rest.
When the season starts, zones along any of the ponds on the map will have "fishing (X)". Then one of my fisherdwarves will try to use the fishing zone, and the instant the job finishes, I get "there is nothing to catch in the (direction) swamps", my fisherdwarf will be empty-handed, and every single one of those zones will suddenly say "fishing (0)". Even when I tried laying a fishing zone on a pond my dwarves haven't been within five screens of, and that said start-of-the-season fisherdwarf didn't touch, the same thing happens to its info.
I saw on some post or another (I don't really recall where...maybe on the wiki discussion page) somebody thinking it might be biome-wide. That is, if a pond in Biome F1 on the map is fished out, then all the ponds in Biome F1 will be fished out (and the same for F2 emptying the rest of F2, etc.). That would explain everything registering as empty at the same time, but doesn't really help when my weaponsmith starts screaming for the shells that I ran out of seven years ago*.
*I'd still have a couple dozen shell left if not for the fact that I didn't realize that shells in an outdoor refuse stockpile disappear with the automatic seasonal cleanup. I almost reported it as a bug that my shells that were safely in a stockpile, unused, and even
forbidden were somehow vanishing into thin air. But hey, you live and learn, and losing is fun! Also, shooting berserk children to death is good practice for marksdwarves
Addendum: And before somebody mentions it, none of the caravans offer anything with shells.