The conflict between France and Britain during this time strongly impacted the political structure of the US. It helped to differentiate one party from the other and it ended up leading to the destruction of the Federalist Party and the onset of the "Era of Good Feelings."
When France and Britain came into conflict with one another, the two parties in the US took sides. The Federalists sided with the British and the Democratic-Republicans with the French. So that helped to structure the US political party system by giving the parties another way to be different from one another.
But then the US ended up fighting the British in the War of 1812. This led to the disintegration of the Federalists. This was because they were opposed to the war and called the Hartford Convention to discuss opposition to the war and even secession from the US. When the war was won, the Federalists looked like traitors and lost the last remnants of their support.