Hello, Palin. As a native English speaker I would ask/say: Does anyone know whose dog this is? You could also ask Does anyone know the owner of this dog? Another option. Does anyone know this dog's owner? I think the first option is the best. I hope this helps you. 😀😉😀

They are all good except, ' Does anyone know who has this dog? ' means something different. If you found the dog, then you have the dog, for now at least.