You could argue that the spinning rotissery is the main factor in this dish, it is in the name in most cultures. The type of meat used differ indeed, but most recipes I’ve read of doner kebap do include a spice marinade f.i.:
Combine flour, oregano, salt, garlic powder, onion powder, Italian seasoning, black pepper, and cayenne pepper in a large bowl.
But the main thing I was going about is that the two nations share a bunch of history, and therefore culture, but after Greek independance they have tried very hard to distance themselves from each other. The main difference is that in Greece you’ll never find an option for hot sauce with your gyros, whilst in Turkey it is on offer.