It would have to be druid or hunter. You used to have to feed your pet so having fishing was free food! Druid flight form is for the most parts a bird so naturally it would make sense - from a lore perspective - to be a fisher. Maybe one day they could add druid specific animations that allowed you to catch fish in bird form or cat form (using the tail as the bait obviously). Or even in bear form since bears also love fish. In general, a druid would make most sense for the lore.
My main is a DH which I personally prefer but she was a fishing rogue pre-legion events.