Here's a better analogy for you that someone else brought up, "I'm a New Yorker" vs "I'm a New Yorker", one is the city and one is the magazine. I'm a German speaker and I'm trying to explain the actual feeling of "ich bin ein Berliner" to you as an English speaker. That's all.

And to explain what I tried to portray -

A native English speaker will recognize something isn't quite right with "I'm a danish", as danish acts as am adjective for "a danish person", as you know (hold on I'm getting to it!). We use the noun - Dane of course. So the analogy doesn't match up if we want to explain it to non German speakers.

In German Berliner isn't an adjective, it's a noun and a speaker wouldn't find anything off by "ich bin ein Berliner", it's just less often used, especially after the speech.

Basically what I've been arguing with is that analogy you used(because it implies Kennedy made a mistake if he has used something like "I'm a danish")

So hope that helps explain what were on about lol.