Fish keeping is more of a hobby than just another kind of pet

I definitely agree that keeping fish is a hobby, rather than having a pet. Where a pet usually enjoys being around you, and interacting with their human, the fish just swim around in their fishtank, basically unknowing, and uncaring, about the life of the human that feeds and cares for them.
Even raising birds, like parakeets or canaries, is closer to having a pet than raising fish is. The parakeets are very intelligent, and they actually enjoy being around people; even though they don't cuddle up on our lap, beg for our food, or sleep with us like our dogs and cats do.
I think that the closest that fish actually come to being considered a pet, would be the Koi, who can be trained and actually do come right up to interact with people who are feeding them in their ponds.