The United States is a representative democracy, and there is nothing in the U.S. Constitution that requires candidates for president to belong to any political party. The people decide.
For much of U.S. history, there were more than two major political parties, and that could emerge again. The GOP, as we now see it in the U.S. House, is hardly functioning as a coherent political party. That said, there are large barriers to entry for new parties or independent candidates.
Under our system, electoral votes are awarded winner-take-all to the candidate who gets the most votes in each state, with the exception of Maine and Nebraska. That means there is no prize for second place. So unless you can actually win a plurality in the state, your impact on a race is likely to be just pulling votes away from the other candidates. Now, that could change at any point. Kennedy and West are trying. But opinion polls so far do not show them with a clear shot at winning any state.
To win the presidency, the candidate needs to gain a majority of the electoral votes — 270 or more. If no candidate wins 270 votes, either because of a tie or because three or more candidates split the vote, the 2025 House of Representatives will decide the president, with each state delegation getting a single vote. Republicans are expected to retain their advantage in the number of state delegations they control.
So until there is a popular rejection of the two major parties, or an official divide in one of them, the two-party system dominates. For more on third parties, read the transcript of my chat here.