Fishing is different from the other crafts. The level cap of the fish doesn't restrict whether or not you can catch it; unlike the other crafts where there is a limit to how high over your cap you can go before it tells you the synth is too high. Personally, I've gone to level 30 fishing skill on my old character just fishing in Sandy for carps.
On a side note, I highly suggest not fishing for skill ups unless you have an ocean of patience. It will drive you less crazy to target gil earning fish and fish needed for other crafts like cooking.
Whilst helping a friend with the SJ quest.
Ranzin: You're killing them all to extinction.
Fenneshyde: Yeah, I'm good at extincting stuff.
Fenneshyde: It's what I'm best at.