How is political ideology used to define who is and who isn't American?
I would argue that we define who is and is not American, at least in some ways, by the political ideology that people hold. This is a much more important determinant of who is and is not an American than anything like race or class. We believe that Americans have to believe in republicanism (which is more or less another term for democracy).
This can be seen most clearly when you look at the sorts of political ideologies that are abhorrent to Americans. Many, if not most, people would argue that a socialist or communist cannot be an American because those ideologies are antithetical to American republican values. The same would go for someone who was a monarchist.
We in the US define ourselves by our belief in democracy. People who are not committed to that set of values are not typically seen as authentic Americans.