I think Rumbo is technically a known stray, not an actual pet, but I am still putting yes, because two men seem to care for him enough to feed him and make name tags for him. Both even protect him from Ron Perlman’s character before Rumbo meets his unfortunate end during his rescue of the animals in the testing facility.
Animal control officers handle puppies really aggressively; Fluke is tied up outside during a thunderstorm with no cover or protection at the junkyard; an animal testing facility is shown with rats, a monkey, cats, dogs, and pigeons, and Fluke has eye drop tests done on him.
The events of the movie begin with a husband/father being killed in a car crash and reincarnated as a dog. Another death is alluded to when Rumbo reveals that in a past life he was the brother of a man he has since visited regularly as a dog.
Eye drops are tested on Fluke when he is held at the animal testing clinic. He is temporarily blinded, and needs Rumbo to lead him out of the facility.
Tear-jerking, but overall optimistic. Fluke chooses to exile himself from his family so that they can be allowed to move on with their lives, and is reunited with an old friend.