Your body's defenses don't work properly against foreign blood, and that's the best scenario for a botched transfusion. You knew that blood types have to be compatible, but did you know that you also have to check for allergies? "I once had a lady that had a bad allergic reaction after receiving plasma. It turns out she had a peanut allergy and the donor had ingested a PBJ prior to donating, which caused the patient to go into anaphylactic shock." Note: If that sounds like bullshit, it's not.
That's one of many reasons finding the right blood to transfuse is much more difficult than ensuring that you've selected the right type -- although that's tricky too, because there are far more types than you were taught, and some are all but impossible to find. "There are over 30 other blood groups other than your standard A, B, O, AB. There is even a blood type called 'Bombay' which naturally makes an antibody for every other blood type. People with this type cannot get blood from anyone except from others with Bombay. It's extremely rare -- about four in a million."
Giving your magic blood an ounce-for-ounce value somewhere between liquid gold and printer ink.
Yeah, if you lost that genetic lottery, don't expect the hospital to have any spare blood for you when you accidentally run your arm through a wheat thresher. You would think this would mean they would take any and all donors to cast as wide a net as possible, but ...