Yes, blood has DNA, and for a time after you receive a transfusion, both your DNA as well as that of the donor(s) would be detected if a DNA test was done. That has no impact on the DNA that makes "you" who you are - it is just in the cells and proteins circulating in your blood. The DNA in the cells in your bone marrow and organs are not affected in any way by a transfusion. New blood cells are produced in the liver and bone marrow, and they will contain your DNA. The cells from the donor will eventually die and be removed from your bloodstream.
A bone marrow transplant, on the other hand, will permanently add the donor's DNA to your blood - but ONLy to your blood cells.
