I would think that it be best to notify the prior owner for them to come get the pet. If they dont, I would think its your pet if you want it.
I would guess it be like lost and found property.
If you lost cats..I generally say forget about it. Foxes, owls, coyotes, coons, and whatever can get them.
You’re assuming it’s the prior owner. Dog could have changed hands 5 times.
You bring it to the pound....not a vet. A vet’s job is “care” not a lost and found operation.