Both Döner and Kebab are words that passed into English and other European languages from Turkish. Importing these words to form an ungrammatical phrase is a feature of borrowing words from another language. While the new word, and new food, may be considered a word of the importing language, as many English and German words are, they are never considered the origin or birthplace. Same goes for food.
With this logic of changing something on top of the same base thing a calling it originating in a new country already shows itself as contract manufacturing, and many would considered slapping a Made in the U.S. label while all the work except a laser logo engraving comes from somewhere else a malpractice and marketing customarily, although it is legal.
With the same logic, one can even go as much as culture-stealing with calling all the damaged cultural heritage in the British museums a British artifact, since they are no longer the same artifact they were in their homelands. Hell, lets go even painting these old statues with modern paint practices and call them originating from wherever they are painted.
Origin is something, cultural assimilation in a neutral connotation is another.