Over the years I've kept pond fish that include ornamental carp, grass carp and even trout.
Apart from the ornamental carp, these were really wild fish that were kept in large outdoor ponds. However, the carp are very intelligent and affectionate fish and when you feed them they will come to the surface to greet you. They will even poke their heads out of the water to eat from your hand.
The trout are much wilder, and though they will approach the bank at feeding time they're not like the carp.
Personally I would consider the carp as pets and the trout as not.
What about others here, would you consider fish that interacted with you, even if they are not in the house and live most of their lives like wild animals as pets?