My previous answer was to the original question which focused on red blood cells which do not contain DNA. Now the question has been revised to focus on tissue transplantation, where the cells do contain foreign DNA, my amended answer deals with a couple of the points raised, but only briefly with the question of identification.
Incorporation of foreign DNA into the human genome
It is extremely unlikely that any of the DNA from the tissue of the donor would be incorporated into the DNA of the cells of the recipient. The donor cells would have to be broken (i.e. die) to release the DNA, which is most likely to be degraded in the process. Even if it survived it would then have to be taken up by the host cells, which is unlikely as there is no specific mechanism to allow this. However if foreign DNA somehow got to the nucleus it would need specific machinery to be incorporated into the host DNA. This involves enzymes making staggered cuts in the foreign and host DNA to allow annealing and enzymes to ligate the two together. We know about this sort of thing because retroviruses and viruses that can exist in the genome in a latent state have specific mechanisms for doing this. It is unlikely to happen just by chance.
Even if foreign DNA did get incorporated, it still has to be in an environment that allows it to be expressed (appropriate promoters). A more likely possibility is insertion in a position that affects a host gene. This can result in uncontrolled division of cells, i.e. they become cancerous, and is a problem that has been encountered using viruses of the type mentioned to insert genes into genomes in gene therapy.
Do the differences in the DNA of transplanted tissue matter?
Obviously differences in the DNA of different individuals has an effect on the individual. It is genetic differences that cause people to differ in size, shape, features, hair colour etc. Oh, and some are girls and some are boys. But when the tissues and organs have finished developing it probably doesn't matter if a piece of transplanted liver is from a male or female donor. The genetic programme that causes sexual differentiation and function isn't the one being expressed in the adult liver. The genes that are being expressed are those for the enzymes the liver needs to play its metabolic role (too numerous to list), and if they work, they work, even if the proteins differ in amino acid sequence by the odd amino acid that doesn’t impair its function.
The problem of identification by DNA
I would have thought that this would not really be a problem. After all, if you want to take a DNA sample from a person you can do it with hair roots, white blood cells or scrapings from mouth skin. You wouldn't need to resort to liver biopsies and the like.
Doctor of Philosophy
Finally, I can see there is a general philosophical concern over who one is if one has tissue or organs from another donor, but until we have brain transplantation I’m not going to worry about that. (And when that comes, I won’t be answering questions on the subject.)