Its akin to natural transplantation.
If you recieved a heart from a doner would you be your doner or would it simply be their heart in your body pumping your blood?
Goes back to the Greek Ship of Theseus. The ship was at sea so long that the individual boards of the ship started have to be replaced, and, they were replaced one by one, until all the original bits had been replaced
The owner of the shipyard that had done the work took all the older boards, including the old nameplate, and built another ship, with the same name.
Which is the real Ship of Theseus? The original, or, the one that the shipwright build out of the old parts.
The incorporation of the twin's DNA happens when the embryo is BB size, or smaller. Not a heart, but cells that develop into a heart later.