Presumably you'd need to be a true bilingual to tell if they feel the same, and even then, your thought processes aren't going to be the same as either type of monolingual.
Yes, that's a very good analogy.
If you say "Ich bin Berliner", it just means that you live in Berlin. I doesn't put any emphasis on identifying with a certain group of people. It merely says that you belong to this group, maybe just by accident.
However, if you say "Ich bin ein Berliner", especially in the context of his speech, it means that you identify yourself with the group, i.e. with the people of Berlin.
That's why I disagree with that point of the Wikipedia article. "Ich bin ein Berliner" was a perfect formulation. I guess Kennedy got the help of a native German speaker or had a very good translator.
Either that, or it was the usual cause for rumors: sciolism.
It could also have started as a dumb joke, which was then taken seriously by people who didn't know better.