(That's irrelevant. However, it's also an urban legend. Be rest assured that no one in Kennedy's German audience was even for a second confused and didn't understand exactly what he meant. He was perfectly clear. Even if you argue that his choice of words is grammatically questionable – something I disagree with and I think it works quite well as an emphasis and makes it clear that he is speaking figuratively and not simply declaring that he is from Berlin or was born in Berlin – the context of his speech and the preceding sentence as well as the sentence itself make it crystal clear what he is talking about and also insures that no one is thinking of baked things.)

I'm French. There is no hidden pun (at least perceptible by French speakers), sorry.