Why can they call it the "American Diabetes Association" is a pretty easy question. The First Amendment.
Part of the First is that the government cannot ban you from using a word unless there is a very good reason do so. "American" is a word that is not associated exclusively with the government, nor was it ever used that way historically. In fact, the word "American" predates the US Government.
Since the word does not now, nor has it ever, implied governmental affiliation, it would be illegal for the US Government to attempt to obtain a monopoly on the word by banning others from using it.
Part of the First is that the government cannot ban you from using a word unless there is a very good reason do so. "American" is a word that is not associated exclusively with the government, nor was it ever used that way historically. In fact, the word "American" predates the US Government.
Since the word does not now, nor has it ever, implied governmental affiliation, it would be illegal for the US Government to attempt to obtain a monopoly on the word by banning others from using it.