I'm Canadian, and I'm afraid I'm going to have to disagree with the premise of this game.
"America" is not a continent. Even if you accept the point the game is trying to make, it is *two* continents. In English, those two continents are generally referred to as "The Americas", and never as "America". We like to think of continents as contiguous landmasses, but that simply isn't how we define them; Europe is connected to Asia, but it is not the continent of Eurasia, nor is the United Kingdom its own continent. Even if we did use that definition, the Panama Canal finalized that debate with nuclear bombs.
That may not be the case in other languages, but to be clear, what you are making is simply a semantic argument. "We are all Americans" may sound like a message of unity, but all it is is arguing over the definition of a word; in the end, it is a pointless position to take. Language is defined by its use, not logic.
And if we did, for some unfathomable reason, choose to call the two continents of North and South America "America", what precisely would be the word to refer to people whose nation of birth is the United States of America? United Statians? U. S. Americans? "Americans" suddenly isn't precise enough, and you don't have an alternative word for it.
Too long, didn't abbreviate: I'm not American.