Thousands of years ago the US had a thriving civilization of many Native American tribes, but the modern American sees themselves more as a successor to Europe than to Native American culture.
Since Egyptians nowadays speak Arabic, and was invaded both by Greeks and Arabs, would modern-day Egyptians see themselves as continuing an Ancient Egyptian culture (of pharos and mummies), or the Arabian/Islamic culture?
I don't see it very well elucidated or widely discussed in near-contemporary Egyptian politics but I think Nasser's three circles theory is worth discussing here. The idea behind the three circles is that Egypts unique position is at the intersection of an African identity, an Islamic identity and an Arab identity.
Without getting into 20 year stuff (and this has been incredibly controversial in the past five years so a full discussion of what contemporary Egyptians think would definitely require that) we can look at the Egyptian constitution under Sadat as a reflection of this, especially articles 1 and 2 which say that "The Egyptian people are part of the Arab nation" and "Islam is the religion of the state and Arabic it's official language."
That being said, the pride that Egyptains feel in the Ancient civilization of Egypt is unmistakeable. Al-Ahram, "the Pyramids", is Egypts national newspaper of record, "Egypt is the mother of the world" is the cliched phrase that every Egyptian proudly extolls, ancient Egyptian iconography is everywhere and Ancient Egyptian tourism accounts for a huge percentage of the national economy.
That being said, I think it would be hard to say that contemporary Egyptians view themselves as a continuity with that tradition. That's in large part because that tradition has a lot of negative connotations to it, particularly the autocracy of Pharoah, which I think has been discussed elsewhere on /r/askhistorians as one of the pre-modern examples of who was rhetorically "literally hitler" before Hitler. That analogy to the pharaohs in a negative sense continues in Egypt to this day.
How about genetically? It seems like the United States was an exception in its near-annihilation of the native genotype, helped by relatively exotic diseases like smallpox and sufficiently disparate technology(e.g. weaponry). More similarly developed neighbors invading each other in europe/asia/africa wouldn't be able to wipe out and replace an entire population like that would they? Culturally, sure, a group could get completely assimilated over time and through aggressive administration, but that's a LOT of people to wipe out "manually", so wouldn't modern Egyptians still be the genetic descendants of the ancients?
Of course,
Mate I'm an Egyptian
In Egypt we just passed the Arab idea from our mind however we are proud of our grandfathers and our race also we are proud of our Arabic identity,
At the past the Egyptian people thought that saying they aren't Arabs means that they aren't Muslims
But prophet Muhammad himself said the Egyptian aren't Arabs but they are copts (Pharaoh)
Also he said that the copt will lead the Islam some day he said ( god, god ! on the copts, you will show up on them and they will be the help and power in god's way)
It happened many times, and still happens tell now
You can define us as Muslim Egyptian Pharaoh with an Arab identity , Also apo blood test proves that the current Egyptian are the decent of their grandfathers