The Second American revolution as actually a fairly confused time. It started out with Reds fighting to restore the structure of the USA after the White Junta had violated several different parts of the 1787 Constitution in order to keep the Communists from taking over via the Ballot Box.
It was only decided near the end of the fight that the Second Civil War would be a Second Revolution, instead of simply a restoration of the old USA.
Even then, the Basic Law that changed the name was composed by a convention called by WCP controlled State Legislatures and passed as an amendment to the 1787 Constitution.
So what if America had kept calling itself the USA instead of switching to UASR?
Note: To those of you who say "state" would not have been kept under marxist doctrine for the Republics/Commonwealths, it was not inevitable. In his writings on the First American Civil War, Karl Marx consistently called the states "states", so there's enough precedent for not changing the nomenclature.
OOC: This takes place in Jello Biafra's Reds: A Revolutionary Timeline