No, they used some of the names from the Roman system so it wouldn't be as obvious that they were just copying the British system. The roman system was completely different. The roman system didn't have a "lower house". Senators served for life. Rome had praetors, consuls and censors. There's nothing equivalent to any of those roles in the US system. The US doesn't elect army generals which was a key part of the Roman system. Romans voted on laws directly, not using the representative democracy that the UK and US use.

The US senate is basically the "house of lords". The house of representatives is basically the "house of commons". The king was renamed to a president, but has many of the same powers. The supreme court follows roughly the same model as the British one. They eliminated the role of the prime minister, but then just recreated it as the speaker.

The main difference is that the US system is more closely based on an earlier British system when the king had more power. In the modern system the monarch is a figurehead who basically only has ceremonial powers and it's the Prime Minister who's really in charge. In the US system, the "head of state" has all the earlier kingly powers of the earlier British system, except that it's an elected position and not hereditary.