Not sure which countries you are referring to ChickenHouse, but not true in the Western World.
A lot of human tissue is transported on commercial flights with a volunteer courier. The Charity or organisation overseeing it provides a simple letter but no real permits or licenses.
Kidneys have a shorter sell by date and are usually coordinated via National Transplantation Systems that liase with each other to ensure the best recipient tissue typing. They then use a number of turboprops and jets which may be little more than a one aircraft operation, although not infrequently they charter in the aircraft. They dont use private pilots simply because they need reliability
Hearts and lungs need to be moved quickly and a surgical team normally goes from the recipient hospital to harvest the organs. They charter an aircraft for longer distances, and use road vehicles for shorter distances. The aircraft are again from a private organisation that is known to be able to deliver and big enough to move a team of perhaps four.
So if a small aircraft is needed it will be for an organ where the overall health cost of the transplant procedure means the aircraft cost is relatively insignificant. Although there is nothing stopping the use of some PPL in a 152, I cant really see the advantage. Certainly a surgical team wont agree to that sort of arrangement!