I made the automatic assumption the poster meant a cat that was well domiciled and enjoyed human contact. Come cats that are pets exist in a more co-living agreement, where you just agree to live alongside them and you keep your lives separate. Very much a cat person myself.

What exactly is confusing about the term "pet cat?" It's a cat. It's the man's pet.

"he has a pet bird" "she has a pet squirrel"

There's nothing wrong with "pet X", you're just used to dogs and cats being pets by default.

Barn cats, mousers