I know the question of what race where ancient Egyptians gets asked a lot, but it almost always goes in the direction of race wasn't an ancient Egyptian social construct.
I've been wondering more precisely and have had trouble finding out what race(s)/ethnicities did they physically resemble and were related to especially before they were influenced by Alexander.
Well, race in Ancient Egypt has become a big debate among historians. Some suggest that they were lighter skinned, and others say they were black. In paintings by Egyptians, they appear to be light-brown skinned, but Egyptian paintings have never thrived for realism.
As you said yourself, in Egypt, race was not part of society, so there is not much information from Egypt on race. People from outside of Egypt have described them differently, however. Herodotus described them as melanchroes, meaning "black-skinned." He also compared their skin color to that of the people of the Kingdom of Colchis, a nation in the Southern Caucasus.
In 2005, National Geographic launched an exhibition that featured a reconstructed face of the King Tutankhamun, where some scholars said that the reconstruction made him look "too white," and thought that he should look darker-skinned. (This is what it looked like, by the way.)
Nowadays, the debate is not as prevalent anymore, but it seems that the light-brown skin tone is very accepted.
