Gyros is made from spicy, marinated meat. Döner is made from just meat and fat, without any spicing or marinating or whatever. Gyros is made from pork, döner is a mixture of lamb and beef. They're also served a bit differently.
They're plenty different originally. In Germany they've been bastardized a little bit and brought close to each other. And then Germany went and declared they invented döner kebap, which is of course utter bullshit.
Turkish shawarma doesn't exist. That's more towards the middle east. You won't really, find it in Turkey. Though I wish you could, because more diversity is always more better.
Anyway, the way naming kebap dishes works (kebap is not a dish, it's the name of a large and diverse family of meat dishes, not unlike salad) is you can introduce all sorts of variations into an existing dish, afterwards you're free to slap your own name on it. There are hundreds of examples of this in Turkish cuisine. So, Halifax Donair is fine. You invented a new variation of an existing kebap dish, you get to name it and claim ownership. That's how it is. What Germany has done is put their own regional spin on döner kebap, which had long existed, and then claim to have invented döner kebap itself. Call it Berlin kebap or whatever, but don't use the name of an existing dish. That's like claiming ownership of pizza margherita just because you added a couple new toppings and baked it in a square pan. It's dumb and wrong.