Not really. Keep in mind after the Ancient Egyptians, Egypt was ruled by many empires (Roman, Greek, Islamic, British). Egypt basically only got its independence after thousands of years of foreign rule in the last century.
So it's viewed as part of the country's heritage but not necessarily culture or state.
Another question, then, if I haven't overstayed my welcome.
Did Egyptians consider themselves a distinct nation under thousands of years of foreign rule, or did they consider themselves Romans at some point, then Greeks, then Islamic, etc (obviously not British)? Or did they always consider themselves an independent nation?
Egypt is like India in that any invaders who settled down there assimilated into the local culture pretty soon enough. The Greeks assimilated into the Egyptian kingdom during the reign of Ptolemy. Roman governors did the same, adopting Egyptian customs and of course intermarriage (cue Mark Antony). The Umayyads did not assimilate as much, often forcing the locals to adopt their stricter religious stances (like modern day Islamists). And like that, they suffered from lack of popularity and eventually revolts in Egypt (along with Iraq, Iran and anywhere not in Arabia), which led to the Abbasid Caliphate. The Abbasids ruled from Baghdad, but Cairo was given significant importance like an Eastern capital of sorts. When the Abbasids soon began declining due to internal weakness, Egypt gained more importance gradually at the expense of Baghdad. Saladin, a Kurd, ended up ruling from Cairo himself. When Baghdad fell because of the Mongols, Cairo effectively became the preeminent Muslim authority up until the conquest of Constantinople. Ottoman pashas often assimilated into the local culture. In fact, the first ruler of Egypt who fought for its independence from the Ottoman Empire was an Ottoman Pasha himself.
Edit:- also helped massively that Egypt controlled the Suez, and the proximity of Cairo and Alexandria (which Baghdad lacked) meant that the spice trade would inevitably make the Egyptians more prosperous. And of course prosperity leads to autonomy.
So it's viewed as part of the country's heritage but not necessarily culture or state.