...Americans are the only people of the world to call themselves by the name of their continent...
This is a regional thing. It's semantics. Europeans refer to N. America + S. America as 'America'. North Americans don't really do this.
As far as Americans naming themselves after the continent, they really don't. The country is called the United States of America. They name themselves after the country. It's no different from citizens of the Republic of Blank calling themselves Blankers/Blankicans/Blankians.