Warning - this may just be a tad long! Many physicians don't take courses in blood banking so it is not a surprise that he has it all wrong.
The immune system has a concept called self. Self is the combination of every antigen (usually any protein over a specific weight and other stuff as well) that you possess. Anything that is not -self is regarded as foreign and should be eliminated. Methods of elimination include breaking the protein apart, killing an organism, altering the antigen's structure, and making antibodies against the antigen.
All human red cells carry antigens on and within them. For example ABO are antigens. Your ABO group antigens don't cause trouble in you since they fit the profile of self. BUT, there are lots of proteins 'out there' that look like ABO antigens. As a consequence, by the time you are 18 months old (or so), you will have made antibodies to the ABO antigens that you do not have. For example - those of you who are group A possess anti-B and those of you who are group O (which is really zero) possess both anti-A and anti-B.
Rh (or D is you use that naming system) is also an antigen but there are no naturally occurring proteins that look like Rh(D) to your immune system so you do not make antibodies to Rh(D) until and unless you are exposed to Rh(D) antigen via a blood transfusion or pregnancy. Antigen i is found on red cells and a similar protein is found on a bacteria called Mycoplasma pneumoniae. People with this infection can sometimes develop antibodies against the bacteria that will also kill off their own red cells. In addition there are over 30 more red cell antigens which your immune system thinks are ok since you have them.
All human white blood cells and platelets also carry antigens. We use a different naming process for them - the CDs. Plasma can passively carry antigens as well. For example, suppose someone drank a glass of milk prior to donation. The milk proteins are antigens and would be given to someone in the plasma. You can just imagine the reaction you would get from that if you were allergic to milk proteins!
So - suppose you got a transfusion of whole blood. You would be getting the potential of foreign antigens from red cells, white cells and plasma. To lessen that potential, you "draw off" the plasma or you wash the cells so that none of the plasma remains. That only eliminates the PLASMA proteins - not the red cell or white cell or platelet antigens.
So - now that you have washed cells, you remove the platelets and the white cells. There are several ways to eliminate white cells. You need to use at least 2 methods since granulocytes and platelets can be eliminated rather easily but lymphocytes require special handling. One way to get around the lymphocyte issue is to irradiate them, killing them. You want to do this if you suspect that the person may be a candidate for a transplant or already had one since this is the best way to lessen/stop GVH.
This means you are left with those pesky red cell antigens. The only truly compatible red cell donor for you is you. Failing that, you could try your identical (not fraternal) twin. If neither of those options is available- you are stuck with getting red cells that definitely have foreign antigens.
These red cell antigens will vary in their ability to cause you to develop antibodies. 30 ccs (1 shot glass for those of you who don't think in the metric system) of blood that is ABO incompatible will kill. 30 ccs of blood that is Rh (D) incompatible will always cause an antibody response. The others - well they vary. And , frankly so do you. Sometimes your immune system is irritated enough that even a small amount of a foreign antigen will trigger it; sometimes not.
Thus, transfusion are a crap shoot. Everytime you get one you are exposing yourself to foreign antigens and it is up to your immune system's functionality on that day to decide if the antigens are "worth" developing an antibody.