The official name for Mexico is the United Mexican States, a similar name to the United States of America but focused on the region of Mexico, the area which has Mexico City and the State of Mexico.

The problem with applying this to the USA is that, unlike Mexico, the USA was a collection of various colonies with different political and religious traditions coming together rather than the more unified viceroyality of Mexico.

Using words like Atlantic or Piedmont wouldn’t work well as the words described the colonies as is but the nation was partially rebelling to be able to settle lands on the other side of Appalachian Mountains.

Columbia might work as the word was used as a by label for the USA, so the United States of Columbia. This would likely cause Gran Colombia to choose a different name for their country.

Outside of Columbia, I can’t think of a name which would act as a label in which all colonies would agree.