The terms America and Americans have been used for hundreds of years, not just the last decade or two. I've had this same conversation with Mexicans and people from South America. The U.S. was the first independent country in the Americas
recognized by foreign countries and remained the only one for many years. The term American had been used by the British for years to refer to the colonies, and was adopted by the country itself.
In his farewell address, George Washington said this: "The name of American, which belongs to you in your national capacity, must always exalt the just pride of patriotism more than any appellation."
More than that, this is just what citizens of the U.S. are called. New Zealanders are Kiwis and Brits are Brits and Canadians are Canadians. Every nationality has a name, this is ours. The continents shouldn't be called America anyways, as Amerigo Vespuci had little to do with its discovery or exploration, but that's the way it is.
In other words, this isn't some evil American power grab, and an insult to the rest of our Western Hemisphere neighbors, but a quirk of history that won't change any time soon.