I believe they are referring to the fact America is inclusive of the entire north and south continent, not just the United States.

They are saying that the people residing in the USA should not be called americans because americans are not the only people in north and south america

I make a concerted effort to use the full country's name (well, the initialism of its full name): USA. It's not "US", because there are many countries that use the words united states in their name (including the USA's southern neighbor, United Mexican States). It's also not "America", because every country in either of the NA or SA continents are "American".

It seems banal, and making a scene over it is not useful, but it does matter in principle. It's the same reason I try to use PRC instead of China (when the simple act of doing so would not invite antagonism), and DPRK instead of North Korea.

Edit to add: "Usonian" is a term to refer to people from the USA. (It's not the only term, there are several that get used)

Who calls a canadian an american? I would understand you not caring, though, since you are an anglophone.