This question is less simple than you think. What does civilization mean? Cleopatra ruled over Egypt, yes, but she was a Greek, descended from Ptolemy, who served under Alexander. Would you consider her a member of the same civilization as Khufu?

What exactly is the civilization you're referring to as existing today?

When you ask about oldest civilization, what are you asking? First people to develop agriculture? First to build permanent structures? First to develop writing? First to build cities? First to unite a territory of a particular size? First to establish trade?

Would you characterize civilization by the technological epoch? Stone Age, Copper Age, Bronze Age, Iron Age, Atomic Age? How valuable are these terms in describing a historical continuity? What is more useful?

Egypt is more of an ethnic group than a single continuous situation. It can be compared to China in that, though that group is always there, different civilizations (in Egypt's case sometimes domestic, sometimes foreign) have ruled there at different times. Modern Egyptian culture is also very syncretic, as a result of the various dominant groups since antiquity.

I don't know all the Egyptian states dating back to the first settlements, but to give some context for the example of Cleopatra and the pyramids: Cleopatra lived in early AD, when Egypt was ruled by Rome, while the pyramids were built long before that by independent Egyptian Pharaohs of the Iron Age. After Roman rule ended Egypt was one of the early areas conquered by the Umayyads, and remained under Muslim rule for most of its history up into modern times. After the Caliphate fell apart Egypt became independent again, and eventually the Ayyubid and then the Mamluk sultanate formed, which at its height ruled Egypt, the Levant, and a good deal of the Arabian peninsula (including, IIRC, Mecca and Medina), until it was conquered by the Ottomans in the 16^th century. After Napoleon tried (and failed) to take over Egypt at the turn of the 19th century Muhammad Ali Pasha established a (sort of) independent dynasty from the Ottomans. This new Egypt ended up falling heavily under British influence, and in 1914 when WWI kicked off, was outright declared a British protectorate. It wasn't until the revolution in 1952 that British occupation ended totally, and modern Egypt was established. It's gone through some changes since then (it was even part of a brief lived Arab union with Syria, and I think a couple of other countries), but to answer your question the Egypt we know as a country today pretty much dates back to the 1950s, while the Egyptians as a people are much, much older.

Edit: fixed some errors that were pointed out.