For much of the Nineteenth and Twentieth centuries, Greek identity was a tug of war between a Romaic and a Hellenic construct, between an identification with Ancient Greece via Western Europe (or vice versa), and the folk culture informed by the Byzantine and Ottoman Empires.
The Hellenes have won, but that victory is fairly recent. I don’t believe it truly predates the European Union. And that victory has certainly not been as thorough-going as people like to think.
Turks have been the Other for Greeks too long for them to identify with the Turks. Especially when Greeks have so much invested in identifying as European.
All I can say is, there is a joy of recognition when I talk to Turks, that I don’t feel talking to Germans. That’s not just because of the commonalities in low rather than high culture. It’s also because those commonalities have been deprecated in official discourse. They have not been ignored, but they have been cast as something to be embarrassed about. So when I do encounter those commonalities, they are all the more resonant for me.