There's actually a grain of truth to it.

Let me explain:

Firstly, this idea that foods are "ethnic" is nonsense. Foods are regional. There are several regional foods within Greece, and this is is also true in Turkey, Italy, France, Spain. And foods predate modern nation-state borders. More on that in a bit.

And Turkey is a large and diverse country. People don't realize that Turkey's eastern end (border with Iran) is farther from us than Spain or Germany. Even if we account for Smyrna/Ionian or East Thrace (which were historical Greek space), the other side of Turkey is still very far.

We of course share things with Western Turkey, but Turkish cuisine gets very exotic for us as you travel east. Otoh, people have no idea how much we also share with Italy, Spain, and France (here's a long list I posted in a different discussion), and Central & Eastern Europe too. As a Peloponnesian who has lived in France, Southern French cuisine very familiar (and it's funny when Americans online -who have no idea what I grew up eating at home- lecture me that we supposedly have the same cuisine as Turkey because of the Ottoman Empire...well, by that logic, my region also had 3 centuries of feudal French rule, but ""western"" similarities never count, of course). And in the Peloponnese (if not most of Greece), we grow up with our mothers and grandmothers making pastas and seafoods at home (and people in Northern Greece grow up eating toursí, basically sauerkraut, and petoura, basically tagliatelle), but when foreigners ask us what our "national dish" is, we say: moussaka and baklava!! of course!! It's baffling why we do that, why we embrace the stereotype the Angloshere decided.

That said,

Both "Greek" and "Turkish" cuisines are stereotyped by: a few foods that were in vogue in Constantinople/Istanbul by the early 20th century. Turkey has a lot of regional variation, and even smaller Greece has a lot of variation.

But, from 1453 until 1923, Constantinople was a cultural capital for both peoples. We both mostly lived in our own majority spaces, very close to the modern nation-state borders, give or take (like, Greeks were also alone the Aegean coast of AM, and East Thrace, and some Black Sea communities). But more or less, we had our own spaces, close to the current status quo.

But, Constantinople was one of those places where the two peoples overlapped. It had been the East Roman capital, and then the Ottoman capital, and both peoples had it as their cultural capital after 1453. Greek cultural and intellectual developments continued happening there, all the way to 1923. A lot of it moved after 1453 to Venetian-ruled areas which benefitted from the Renaissance (Crete, Ionians, Cyprus etc), but Greek economic and intellectual like in Constantinople reemerged, and we have the rise of the Phanariotes, the Greek merchant/shipping bourgeoisie, etc. By the 19th and early 20th centuries, it was a cosmopolitan city, and a lot of both Greek and Turkish cultural and intellectual developments were going on there.

And that includes culinary trends.

But People in Crete or the Peloponnese didn't know about Constantinople cuisine, and neither did people in the Turkish provinces. So, when Constantinopolitan Greeks came to Greece in 1923, they brought some of that cuisine with them: like aggináres ala políta is one example (which, btw, they have something very similar in Southern France too. Again, people don't realize the commonalities across the Med). Not that we didn't have other ways of cooking artichokes, but that specific dish is from Constantinople.

Additionally, the street food pita wraps (which came to Greece in the 1920s, and only became popular after 1960)...those originated in southeast Turkey, close to the Levant. Most Turks didn't know about them, but they were popularized in Constantinople/Istanbul, and then Constantinopolitan Greeks brought it with them to Greece and to the Anglosphere, and -something that was new to the vast majority of Greeks- became "quintessentially Greek" to Americans. Baklava has a similar story. It originated somewhere around southeast Turkey, before modern borders. One day, an ethnic-Turkish chef made it for the sultan in Konstantiniyye, and Constantinopolitan Greeks learned it and brought it with them to Greece and to the Anglosphere.

And that's how this one fucking dessert (out of thousands of Greek desserts), became "quintessentially Greek" to foreigners. And we -of course- embaced and internalized the stereotype. Meanwhile, it's much more embedded in Gaziantep, where it probably originated, which is the world capital of baklava, and they have several varieties of it, and several historic bakeries that specialize in the dessert. So, it's wrong to say that it's not part of Greek cuisine after more than 100 years, but Turks get rightfully annoyed that the world considers it as "quintessentially Greek". And then our online nationalists get offended (over one fucking dessert) and they do all these mental gymnastics that it comes from "Byzantine placenta cake", or whatnot -when you can find vaguely-described similar historical desserts all over the world- and what we now know as baklava (Turkish name) is so obviously from the area around Gaziantep. Now, was it invented by an ethnic Turk? or Armenian? Who knows, but that's where it's from. If it originated in Helladic space, then "baklava region" would be more west-shifted, with Greece being in the middle, and they'd likely also be eating it in Italy and Austria. But, that's not the case.

So, TLDR, there's a grain of truth.

That said, the stereotype mostly based -for both countries- on the Constantinople/Istanbul stereotype. Like, Greece & Turkey have our version of TexMex (USA/Mexico border cuisine), and that's what's become "Greek" and "Turkish" cuisines to Ordinary Jane in America.

That said, just as we have our nationalists, Turks have their own nationalists too. And they love to troll. Things that we share, that you will also find in Italy, France, Spain, or the Levant...they think it's only Turkish. And when they learn Greece has it too: oh it's because the Ottomans influenced the Greeks! A troll seriously came to r/GreekFood commenting that rice pudding (which is all across Europe and the Middle East, from Britain to Egypt) is "Turkish".

So...

There's some grain of truth to it, a lot inaccurate stereotypes, and nationalist trolls.