“Independent Egypt”? Cleopatra was the last of a royal line that traces its origin to Greek warlords from the 5th century BC. Her bloodline was Greek, not Egyptian.
Yes, but when the Ptolemy's found themselves with Egypt as booty from the carved up conquests of Alexander the Great, what did they do? Put on the headdress, of course, and declare themselves the new Pharaohs.
Just like the Holy Roman Empire was krauts all the way down, and not Roman at all; you go with the theme.