According to the ideas here goldfish can't be pets either pet "pet goldfish" is commonly excepted.
[/LIST]An animal kept for amusement or companionship.
While you may not want a pet that isn't a companion others like them for the amusement, like watching fish to relax or reptiles, frogs, insects. Pet is simply any animal you keep for selfish reasons(companion,looks, amusements) and has no other workable purpose(food source, work animal).
Not the nicest way to put it but that is pretty much it. To me, a pet is any animal you see frequentley and have an attachment to, wild or tame.