Yes, blood groups are much more complex. First, the "blood groups" we are going to talk about are those on red blood cells (RBCs), not white blood cells (WBCs) or platelets. Both WBCs and platelets have groups too, and WBCs are even more complex than RBC blood groups.
The structures, called antigens, on RBCs that make up the ABO blood group system are actually different sugar groups attached to proteins. If you are missing the enzymes (protein that does work) that puts the different sugar groups on, then you are "O". A, B and AB people have A enzymes and/or B enzymes that make the full structures. In addition to A and B there are actually subgroups, like A1 and A2
The Lewis blood group system is also one where sugar groups determine the type. Lewis has several types as well, with Le(a) and Le(b) being most common. You can be Lewis negative as well: le - pronouced little "LE".
Many of the rest of the blood group antigens are actually different proteins or parts of proteins on red cell membranes. The most commonly known of these types of blood groups are the Rh groups. You may have heard that an Rh+ positive baby may have a serious disease if the mother is Rh-. This occurs when the Rh- (or little d) mom has become immune to Rh+ RBCs from an earlier baby or transfusion. She will recognize the Rh+ cells as foreign and mount an immune response that then affects future Rh+ babies. We have a way of preventing that now. The Rh factor is sometimes also called D (Big D). Also the Rh system has C,c,E,e and other proteins as well. [Pronouced Big C, little c, and so on.]
Same as the ABO, Rh and Lewis systems, we have Duffy, Kell, Lu, MNSs, P, X-linked and many other blood groups. Each of these blood groups is a protein or sugar-linked protein antigen that is found on RBCs. Each of these has a function of its own and all are there because of the genes inherited from our parents. They have various functions for the RBC...tracking in the arteries and veins, tracking the age of the RBC so that the spleen can remove the oldest ones, letting in glucose and calcium, keeping potassium in and sodium out -- all the normal functions of proteins that maintain the life span of the RBC (120 days). (Hemoglobin is totally on the inside of the RBC, thus not involved in blood groups.) The differences in groups come about because of mutations in the genes eons ago that were compatible with normal function, but like differences in hair or eye colour, can be used to identify you. Blood typing for identification is like DNA fingerprinting...in fact, it was what we did prior to DNA fingerprinting, particularly for paternity testing. About 14 different blood types would be determined.
Why are all these important? Knowledge and testing of blood types is extremely important in transfusion medicine. This specialty treats negative mothers who have become immune to their positive babies, and ensures that whenever you need a blood transfusion that the blood is compatible...not just ABO and Rh, but all the others as well. Patients who get lots of transfusions often develop immune responses to other people's blood making it difficult to find compatible donors, and transfusion medicine provides the testing of donors and recipients.
Remember my comments about WBCs? All the same stuff applied to WBCs, too, but with WBCs there are 6 major families of groups with 100s of various types in each. These are the groups that are responsible for compatible or incompatible organ transplanation. The complexity of these families of antigens is why it is so diffucult to find organ matches.
Where ever you may live, you should be able to find a medical laboratory technologist in a local hospital who can tell you more.