People from the US are referred to as Americans not because they chose to be, but because they were called this by the British during the colonial days.
People living in what was to become the United States, for the most part, still saw themselves as Britons in the days leading up to the Declaration of Independence. The word "Americans" was used by the British in a derogatory way, as if to say "no, you're not really British, you're American".
Even after the revolution, there was no sense of national cohesion, and most people would have referred to themselves, and preferred to be referred to as, Virginians, New Yorkers, etc. That's why the country almost fell apart early on, and probably would have without George Washington.
If you had referred to someone from Pennsylvania as an American in 1776, he probably would have been confused or offended or both. It wasn't until well into George Washington's time as president that Americans came to see themselves and refer to themselves as Americans, and this was 15 or 20 years after independence was declared.
Bottom line is, Americans aren't called Americans because they asked to be. It was just something that stuck.