The Ptolemies set up shop in Egypt in the aftermath of the death of Alexander in 323, in a period we call the Hellenistic period (vaguely 323 to the end of the 1st century BCE). The Hellenistic period is marked by an expansion of "Greek" culture to every corner of the Mediterranean and beyond, stretching in many ways as far as Afghanistan. Hellenistic culture was modular and leveling, meaning that there came to be a certain "Greek" substrate to many different places, even places that did not speak Greek: the "modules" included ritual practice based largely on Greek norms, but with as many modifications as desired (and this included basic temple shapes, a basic shared mythographic lexicon, etc); a material culture based on Pan-Mediterranean norms, with artistic styles which tended to spread from centres like Alexandria and Pergamon outward to (eventually) be adopted by a variety of Hellenistic cities; a basic "Greek" system of governance based polis citizenship, a representative council, law courts, and a ruling class; a general leveling of military practice based on the basic infantry phalanx (but with many variations, of course); a relatively consistent farming culture based on the Mediterranean triad of olive, grain, and grape (one can argue whether it was "Greek" or just "Mediterranean"); et cetera.

So in this way, Ptolemaic Egypt was just one of many Hellenistic centers, all of which contributed to and participated in the larger Hellenistic world. The basics of "Greek culture" had expanded from the heartlands of Attica and the Corinthia, not only beyond Macedonia but beyond the borders of what had been considered "Greek" in prior centuries. Hellenistic culture became, for lack of a better term, cosmopolitan, worldly, rather inclusive, and intrinsically attractive to a huge variety of cities and peoples of the ancient Mediterranean. Rome, despite her many quirks, was broadly a Hellenistic city in the 3rd and 2nd centuries; so too were the Etruscan cities, by and large, again despite their idiosyncratic aspects. Carthage was in many aspects a typical Hellenistic city. Masalia, the cities of Tripolitania, the cities of southern Italy and Sicily, the Jews in the Levant, and the many cities of Asia minor also. We talk of "Hellenistic Kingdoms," which were the major players in shaping the political landscape of the Hellenistic world, and also the cultural/artistic landscape. Alexandria was a major Hellenistic capital; so too was Epirus, Macedonia, Antioch, Pergamon. The great cities of the Greek Classical period were relegated during the Hellenistic period, and increasingly became the pawns of the Successor States, as power shifted away from Athens and Corinth and Sparta and Thebes and towards Egypt, the Near East, and outward just more generally.

So in short: the Ptolemies would absolutely have thought of themselves as "Greek," and Alexandria had everything a good Greek city should have and much more. But the concept of what "Greek" meant was shifting and would continue to shift throughout the period as basic Greek forms were adopted and deployed in distant, often strange places. The textbook example are the Greek inscriptions put up by the Greek man Klearchos in Ai-Khanoum (in modern Afghanistan), copies of maxims from the great temple at Delphi. Ai-Khanoum was a distinctly Hellenistic place, with a large theater, shaded colonnades under Corinthian capitals, Greek temples, statues in trendy Greek style, and even a gymnasium.

A go-to source for the Hellenistic Period is Peter Green's monumental Alexander to Actium, which you can pick up used in hardcover for relatively cheap. It's a recommended read.