OK, I'm going to get technical here. According to the American Heritage Dictionary, the definition of a pet is: "An animal kept for amusement or companionship." I don't think too many people consider their fish a companion, but I'd say most fish are kept for their entertainment value.
I take issue with the term amusement. It comes from an Old French word amuser, which is to stare stupidly. It's opposite of the word muse, which is to think deeply or to meditate. I prefer to muse on my fish.