This is a pretty specific question, which is why it includes all of medieval Britain.
I'm aware that farmer would keep cats to get rid of vermin, but would someone own a cat just for companionship? Did cats ever populate monasteries with monks?
Oh my goodness, one of my first AH posts was on, as OP there put it, "crazy cat monks." Although it was definitely not just monks--there are plenty of stories of nuns and cats, and quasi-religious women (who did not take formal monastic vows but lived an especially pious life) and cats.