A pet is Personal property. meaning something that is currently in use by someone due to necessity, both physical and psycological. the house you live in is personal property, the one you rent out is private property. one is necessary for everyone, the other is a tool to extort capital.
Akshually, communism does not ban all kind of property, you still get to keep personal property which only entails the use of goods, it only bans lucrative property ie the deriving of value from owning the good. So puppies can still have owners under communism 🤓
Well... technically, he spoke of private property, not personal possession. So as long as the master doesn't make money by exploiting this lovely doggy... no problem here :3 (be happy !)