Yes. The owner is the client, and the animal is the patient. And the animal doctor is responsible for developing a good veterinarian-client-patient relationship. (I don't know why, but I can't stop myself from laughing as I write this.)

Not according Oxford, a patient cannot be used to refer to a sick animal because a patient is "a person receiving or registered to receive medical treatment." Oxford Dictionary's definition specifically says that a "patient" must be a person. Source: http://oxforddictionaries.com/definition/english/patient