That all-knowing entity Wikipedia puts it best -

“In fact, Kennedy’s statement is both grammatically correct and perfectly idiomatic, and would not be misunderstood in context. The indefinite article ein can be and often is omitted when speaking of an individual’s profession or residence but is necessary when speaking in a figurative sense as Kennedy did. Since the president was not literally from Berlin but only declaring his solidarity with its citizens, “Ich bin Berliner” would not have been correct.”

Furthermore, Germans don’t refer to jelly doughnuts as Berliners. They generally call them either Pfannkuchen or Krapfen.

So there.