[anthroDominator]

The word prison implies punishment, which is not always a component of domination.

Pet is a more open term, and represents the concept of an enjoyable living possession, retained without enslavement.

Enforcing submission may carry the possibility of punishment, but an equitable game can be achieved, even amid power differential.

Is the animal’s environment engaging and enjoyable, yet perhaps limited? Then congratulations, it’s your pet.