There are plenty of American political parties, including one literally called the "American Independent Party," which was founded in 1967 in order to nominate George Wallace to run for president on a segregationist platform after the two big parties were no longer willing to die on that hill.
Usually, if you hear about a politician being an "independent," it means they're either a legislator not siding with either of the two major caucuses or a candidate running without the backing of any party, which does happen sometimes when a candidate is wealthy enough to fund themselves. The last time anyone managed get very far doing that nationally was Ross Perot in 1992, but when he ran again in 1996, he formed his own political party instead.
