The Ottoman Empire had what is called the Millet System. Basically the Greeks were allowed to keep their own culture, customs and religion. In exchange they had to pay a poll tax as well as provide their sons for the Jannisary Corps.

In fact the Greek Orthodox Church was the first non-Muslim church recognized under this system in 1454.

Greece and Turkey have had an enormous amount of cultural exchange over the past half-millennia. Every corner of their cultures has been influenced by one another and the rest of their brother subject nations. They are united to one another by the Ottoman Imperial legacy.

The Ancient Hellenic language spoken by Aristotle and Plato are at least as different from modern Greek as Chaucer is from modern English, including many loanwords from Ottoman Turkish. The Greek language has been influenced by Turkish much like English was influenced by French following the Norman conquest and centuries of Aristocracy speaking French over English.

To a large extent the Turkish inhabitants of modern Turkey are descended from Greeks, or, at least, Greek speaking Orthodox Christians who lived in Anatolia during Byzantine times. Genetic studies do indicate that modern Turks are much more closely related to Greeks than to Central Asian Turkic peoples. So quite a few Greeks did assimilate to Turkish culture and/or intermarry.