Many countries include 'Democratic' or 'People's' in their names, even though they are not democratic and do not serve their people. Examples:
Democratic People's Republic of Korea
Lao People's Democratic Republic
People's Democratic Republic of Algeria
Democratic Republic of the Congo
And German Democratic Republic (East Side) in 1949–1990.
None of these states are democratic, power and wealth are concentrated in the hands of a small ruling elite. Who are they fooling with these names?
I gave you an upvote because while i'm USian, I firmly believe that "American" is a proper term for people of the United States of America.
I also am fine with the folks (mostly Brazilians in my experience) who want to use "American" to refer to people of the Americas (or the continent of America for those that consider it a single continent).
Both work fine.
It's usually pretty darn clear from context which is meant, just like with Georgia (a US state and the English name for a country).