Turkey/Ottomans are kinda their own thing, meaning their civilisation doesn't get fully explained by their current geography.
They rightfully reject the idea that they're a Middle Eastern country and culture. They're a regional outlier, meaning you don't understand much about Turkey if you study their neighbours.
Even though they are close to the Middle East/Levant, they don't have much in common with neighbouring cultures like Syrian, Israeli, or Arab cultures.
They are fundamentally a Turkic culture that transplanted itself from Central Asia to a Middle Eastern - Mediterranean location, on top of a local population "layer" of Anatolians, who were descendents of Mediterranean colonists (Romans, Greeks, even Celts/Galatians).
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duckzilla
When they adopted a religion that was very common in the Middle East (Islamism) they distanced themselves from Europe, which was entirely Christian. Let's not forget that despite those 3 Abrahamic religions having roots in the same region, they did not see themselves as allies or relatives. Muslims call everyone who doesn't share their faith Kafir (infidel) or mushrik (polytheist). Israeli's religion, Mosaicism, also rejected Christianism as a sect whose initiator, Jesus, committed the blasphemy of calling himself the messiah, even though they were otherwise followers of all the orthodox Torah tenets. Even though these three religions appeared in the same region, there was no cosy cohabitation between them. And later, they even provided a cause for faith-based war. When Ottomans were invading the Balkans and Eastern Europe, they were doing it in the name of spreading Allah's calliphate, extending the glory of their god. When Christians launched Crusades to capture Jerusalem, they were doing the same thing, except to serve the objectives of their faith. And both killed their enemies in the name of their religious faiths, calling them infidels or heathen.
That's why I doubt that there is much significance to this argument that, if these three religions appeared in the same region and they played some role in European history, then they should be considered European. There's really no cultural argument for considering Islamism and Mosaicism European religions, because they never formed a majority in any European territory. Their fiefdom was definitely not in Europe, but elsewhere, in the Arab world, in Turkey, in the Middle East. Jews are a special case, because they had to live in diasporas for centuries. But Jews never considered a European country to be their own, the homebase of their religion. Their religious texts dictate that Israel is their holy land and that's why they created a political movement (Zionism) to recover their lands. If they identified as a European religion, they would have never mounted such a massive effort to recover their lands, plot by plot, and build a country in that barren land.
Also, another argument why I wouldn't place much significance on the fact that Christianism appeared in the Middle East is that, in its early phase, they were a sect that were inspired by a previous ascetic movement, the Essenes, whose faith is speculated to trace its roots in - surprise, surprise - Buddhism. Yep, it's possible that this component of Christianism which urged people to focus on the salvation of their soul and purifying themselves from all material concerns came from an ancient streak of Buddhism that was somehow carried over to the Middle East, among people living in the desert. It wouldn't be the first time when an idea that completely takes over a continent comes from another region, where it wasn't even considered the most important.
After all, both communism and capitalism appeared, in their modern forms, in Europe, and both completely changed China's history and the way they organised their society and economy. That doesn't make China European, though, obviously.