Can doctors just give anybody blood they find in their supplies?

No. There are different types of antigens found on blood (A and B) which can only be accepted by people if they don't produce the antibody for it. If you're type A, you can only accept blood from people with type A or type O (O meaning no antigens). Type AB (has both A and B receptors) can accept from either A or B or O. Type O can only accept from type O. There are also other antigens found on blood besides from the ABO group.