I don't know, there seems to be quite some common ground when it comes to food and drink, for example. That's part (and the result) of some shared culture, I'd say. Isn't it nice to realize that your neighbors might be people just like you are, who like to take their coffee in quite a similar way?
Over a million "Greeks" were deported to Greece in the 1920s, despite many speaking only Turkish and many "Turks" were sent back the other way. Not to mention that most of those inhabiting Turkey today - and identifying as Turks - have an awful lot of non-Turkish ancestry. It's not only Greek of course (plenty of Armenian, Kurdish, Arab, Albanian, Circassian, Bulgarian, etc), but Greeks and Greek culture has had a massive imprint on modern Turkey - and Turks have also had an impact on modern Greek culture.
Of course neither side is particularly happy to admit it, but it's very much there.