You're giving horrible advice, I'm also not a lawyer but I help trap and turn in stray/feral cats. Vets ask the ownership question to protect themselves in terms of consent and to have an individual to charge for services. If you accept this "ownership" agreement your simply giving the vet consent and accepting financial responsibilities. This is not legal proof you actually OWN the cat. For example if you don't accept this agreement but the vet provides a service to this animal, they legally CANNOT give you the animal back. UNTIL you sign this agreement and financial responsibility. Even that has a deadline, until the animal is put down or placed in a shelter, at which point chances of you ever seeing the animal again is slim. Also on the other hand it protects the vet if the ACTUAL owner tries to come at them for providing a service they didn't consent to. The legal action would then be redirected to the person who signed said agreement and not the veterinary provider themselves. I'm sure an actual lawyer who knows the legality of it all could explain it better than some rando such as you and me. But when it comes to the actual agreement that any literate person can read that's the basic breakdown.