Now when the US Civil War happened, the idea of referring to the area of the USA as "America" had become ingrained into the public consciousness. So when the seceding states formed their own government, it was the Confederate States of America. If they had won their war, a problem of demonyms emerges (as Turtledove among others has noted). "American" no longer means "someone from the USA". We might end up with awkward workarounds like always having to say "USAmerican" and "CSAmerican"...
But fourscore and seven years before, the name was still young (and officially for a while it was actually the United States of North America) and there remained some discontent over it. Some of the founding fathers disliked it because of the problem of a demonym - "United Statesian" sounded silly and "American" was unpopular because in that era, "an American" usually meant a native Indian.
However, in the end the name prevailed because the others (Freedonia, Columbia, Appalachia, Alleghenia, etc.) were even less popular. Now let's say there was an earlier US Civil War, or more probably the union fell apart due to a failure to ratify the constitution - whatever. My point is, can we have two recognisably US-like nation states (i.e. not regionally focused like a seceding New England) where the people of one are called Americans and the other are known as Columbians or Freedes or Appalachians or whatever?