Functionally, it remains the same.
Prior to the transfusion, tests for cell surface antigens on the donor and recipient red blood cells (RBCs) would be done to ensure no differences are present that might cause a transfusion reaction.
Both the donor blood and the recipient blood have the same oxygen-carrying properties, so from a blood-draw standpoint it all just looks like blood. In most cases, the transfused blood is also a very small part of the body’s total blood volume, and so any differences between it and the rest of the body will be nearly undetectable in your average blood draw. Of course, if there are blood type mismatches, you would see reactions between donor blood and recipient blood, but barring that, everything would be kosher. Finally, the life of red blood cells is about 120 days, and in most transfusions, the transfused blood is completely absent from the system within 60 days.
One interesting side effect can happen in the event of massive transfusion that replaces an incredibly large volume of blood. If the volume is large enough to essentially replace the patient’s own blood, then you can get something called “dilutional thrombocytopenia” since transfused blood has platelets that aren’t fully functional. This can be corrected with further transfusions, but it’s something that might actually show up in a blood draw.
What would happen if a DNA test were performed on your blood after a transfusion?
Nothing special. Tranfused blood (meaning RBCs) doesn’t carry DNA – it is essentially saline and RBCs. Only a minimal amount of liquid (I don’t know the composition) is added to maintain pH and osmolarity so that the RBCs aren’t destroyed.
Blood that is routinely donated is then prepared in a way that removes >99% of white blood cells (the only thing in your blood to contain DNA) and the plasma. So, really, a blood transfusion is usually an erythrocyte transfusion.
However when someone has a bone marrow (or organ) transplant, all of the blood cells created from that point (IE: white blood cells) carry the DNA of the donor, not the recipient. They never really become the recipient’s. Yes, DNA confusion has happened.
What happens if you put the wrong blood type in to someone?
The immune system will recognize it as a foreign object and destroy it, the same way it would an agent of illness like a pathogenic bacteria. Different blood types have different antigen markers on them (which is what the letter and +/- represent), and your body will attack blood that has the wrong markers. Type O blood lacks antigen markers, which is why it’s the universal donor, while AB blood has both making it the universal acceptor (the immune system is used to all the antigen markers).