The way I see it, English people called the place America when they sent colonists here. That gives us the right to call our country America. We also, for lack of a better name, decided to call the continents of the new world North and South America. That was our name for it in our language.
To tell the people of the United States of America that they can’t call themselves American is to deny us the right to name our own country something based on history. It’s not like we are calling ourselves Brazilian or Mexican or whatever.
On the other hand, all the other people living on the continent(s) decided they were going to name the whole thing America too. Fine, whatever. I don’t understand why you have to use our name but I guess it makes it a little less confusing.
Tell us what not to call ourselves? Go take a hike! And get your own name while you’re at it! We have more of a right to call ourselves American than you do.