Döner is a traditional Turkish meal. It is sold as a dish on rice, on a bread, or in a wrap. However, the Döner Kebab as it is known in Germany today originated with immigrants in Berlin.

I don't think it's easy to say what culture it is. The German Döner is still very similar to the traditional meal, but "Eating a Döner" has engrained itself deeply into German culture.

Normally, when eating foreign food, you'd call it by the country it comes from. If you order a burrito, you're "Eating Mexican", if you order a Pasta you're "eating Italian" and if you are having sushi you are "eating Asian". But you don't "eat Turkish" when eating a Döner.

I don't have an Answer, I just thought I'd share my thoughts.