Living animals cannot be owned.
The keeper of an animal should take care of its interests.
From this we can conclude that though one cannot own animals, this does not prohibits keeping animals per se. Rather keeping animals is a kind of guardianship. This is not a real issue with companion animals (usually called pets), but it seems problematic for the economic use of animals. Hence:
The keeper of an animal is entitled the enjoyment of its fruits, provided that such enjoyment does not violate its dignity and its interests.
With fruits are meant the thing produced by animals but are not part of its body. These fruits are not necessarily material goods, but could also be intangible goods. If Alice has a plot of land and Bob has a goat, Alice could pay Bob to have his goat graze her plot.
Cats have their own idea of human-animal relationships.