I never say "I'm going to the United States of America" - I just say "I'm going to the United States", which is why I would expect to count the US here as a two-word answer. That would also align with the Countries of the World quiz answers: all the other answers in this quiz are identical with the Countries of the World quiz, but your USA answer is the only that isn't.
The quiz isn't about what you would use in conversation. It's about the name of the country. and the official name of the country is "united states of america" I think its fine
If it's really about the name of the country, then People's Republic of China, Plurinational State of Bolivia, Islamic Republic of Mauritania and Socialist Republic of Vietnam (and a few others) are missing as well.