Kidney transplantations are common procedures among animals. Cats and dogs very commonly undergo successful kidney transplantation. There is no need for the tissue of the donor to match the recipient in case of animals for kidney transplantation.
Other organs such as heart and liver are a matter of debate as there is the ethical issue of obtaining the organ from a healthy, living animal. Partial liver transplants have been attempted without success. As liver and heart are single organs, a living animal cannot donate them. Unlike in humans, these organs cannot be used once an animal is dead.