Animals learn from humans they are very smart. Once humans started killing animals with weapons history changed and they adapted. Dogs and Cats also act very much like their owner.
I am talking from experience. I actually studied dogs a lot in the past 7 years since I had 5 in total and was always living with cats. Dogs totally behave like the owners.
Of course animals live without property they can not even read and write so how would they claim their property? They have a territory but that is not their property.
Territory and property are not really different. Apes even raid other groups territory to steal resources. Reading and writing are in no way required for property.