I’m pretty sure the DNA similarities come from the Hittites, not the latter day Turks that live there now (though the Turkish invaders surely interbred with the original inhabitants and still have some of the same genes).
The Hitties were rivals of the Egyptians, but were basically the same kinds of people.
Though the "lost" empire of the Kush, was supposedly comprised of black Africans, their extremely brief time ruling over Egypt, did NOT make Egyptians "blacks"! They didn't intermarry and the whole bunch of them DISAPPEARED, shortly after they lost Lower Egypt, which was the ONLY teeny portion that they ever had any hold over.