> entirely different countries currently in the things that matter.

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?

Different countries ... with a common heritage.

Maybe during history lessons you preferred a good beauty sleep instead of paying attention to the class :)

If you don't see a lot of overlap between Greek and Turkish culture ... not sure what to say.

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.

Greeks are Christians, Turks are Muslims. For starters.

That's a bit of a tautology because the population exchange was done on the basis of religion, not language. There were plenty of Greek speaking muslims and Turkish speaking orthodox christians before 1915.

Many parts of Turkey and N Cyprus still speak Greek in secret

That is both a difference and a similarity. Both are way closer to each other than any two random religions on average.

Perhaps you don't realize that the heart of ancient Greece was really in what we now call modern-day turkey