On the whole, modern Egyptians very much identify with ancient Egypt, along with any number of other identities: Arab, Muslim, Coptic, African, whatever. Of course, this depends hugely on when/where/who you are, but I'll commence with the huge generalizations! It's a fairly intuitive thing for most people: you grow up surrounded by the material culture and there are many remnants of ancient Egyptian culture within the country today. I'll share some examples!

First, there's the language. Coptic is the most recent form of the Ancient Egyptian language, spoken in Egypt through the 7th century or so when it was overtaken by Egyptian Arabic. It's very clearly an Egyptian language, with the same core vocabulary as the Demotic phase with Greek loanwords and various evolutions in grammar and pronunciation. Many of the attributes that differentiate Egyptian Arabic from other dialects are arguably from Coptic as well, including loanwords and some grammatical quirks. (Wilson Bishai's Coptic Grammatical Influence on Egyptian Arabic used to be a classic text about this, but it's quite old and I haven't read it in years so I'm not sure how much of it holds true with newer research.) It's still the liturgical language of Egypt's sizeable Christian minority (known as Copts), and there have been sporadic reports of native speakers in a few families in Coptic villages in Upper Egypt.

There are also more general cultural influences. Sebou', a celebration of a new baby that occurs at seven days, is remarkably common across classes, regions, and religions and seems to be depicted in some New Kingdom artwork. (If you're interested, All Things Considered did a brief segment on the Sebou' celebration.) Sham al-Nessim (which is today, coincidentally!) is a beginning of spring festival celebrated much the same way it was when Plutarch wrote about it, with picnics of fermented fish, scallions, and lettuce, plus the modern addition of hard boiled eggs (yes, you'll have the stinkiest breath on God's green earth).

As far as identity issues go, it's not a huge conflict. Most people identify as both Egyptian and Arab, but this is largely contextual: if you're talking about something on a global scale, you're more likely to think of yourself as Arab in that context than if you're talking about Middle East politics. There are some caveats, though: Copts in particular often specifically reject the Arab label for a lot of the cultural reasons I mentioned before, plus a particularly fraught relationship with the Arabization of the country. Particularly religious Muslims may do a similar thing as far as rejecting the pre-Islamic past. In daily language, the word "Arab" is often used in reference to Gulf countries, but it's not necessarily a rejection of the identity itself. It's also a fairly modern construct. The adoption of Arabic and conversion to Islam happened gradually through the 7th and 8th centuries without necessarily altering the community identity for most people on a fundamental level. In its modern form, Arabization is a very modern undertaking with specific political goals: in short, handling colonialism. Modern pan-Arabism in particular was largely a project of the 1950s and the crisis of independence. For more about this, you can check out The Origins of Arab Nationalism and Sacred Language, Ordinary People: Dilemmas of Culture and Politics in Egypt. Again, it's important to emphasize this isn't an absolute truth: Arab as a cultural label is incredibly vague, and really mainly useful in a western context. You can do a similar thing with African identity--Sisi sometimes plays this up in order to improve those regional relationships as well.

I wish I could think of some examples of portrayals of ancient Egypt in Egyptian pop culture/media, but I'm woefully out of touch--hopefully someone else can chime in!

All in all, Egypt is hugely proud of its history and tends to feel very responsible for it. It's important to be skeptical of efforts to divorce ancient and modern Egypt--it's a classic imperialist technique, one that remains oddly acceptable in this region of the world.