Human organ transplants have become a normal medical procedure during the last fifty years. Organ transplants in animals, usually pets, are, on the other hand, quite rare.
According to this article the problem of animal organ transplants is that unlike in the case of human organs, there is no infrastructure for taking organs of animals who died in accidents. Hence the only option for getting animal organs would be to kill another animal.
However, most people would consider killing a healthy animal for its organs as unethical. And hence organ transplants in pets is limited to kidney transplants as most animals have two kidneys and could donate one by life.
Scientists around are busy to grow human organs in the lab. Human skin cells can be turned into pluripotent stem cells and those could be turned into different types of tissue. The next logical step would be create complete organs from these induced pluripotent stem cells.
If it would be possible to make human organs in the lab, then why not for other animals? From a medical perspective there is little difference between humans and other mammals.