In Massachusetts there are three official political parties: Democrats, Republicans, and Libertarians. In addition, there are 29 political designations, not considered official parties but a voter can enroll in any of them and the political designations can field candidates for state offices.
I suspect other states have similar things.
However, since the constitution requires that the president be elected by a majority of the Electoral College, there is an effective limit of two parties. Failure of the EC to achieve a majority leads to throwing the vote to the house for president and the senate for vice president just to confuse things further (per the 12th amendment).
Won’t work. The staff asks you because they’re required to and because it’s also a cognitive health indicator whether you remember it or not.