> And Greece's government was nothing like Rome.
I presume you're referring to the various systems of democracy introduced in different ancient Greek city-states, as opposed to the Roman Republic. Of course, they're different. But it doesn't run counter to what I said - being descendant from the ideas of democracy doesn't promise you that any aspect of the government will be in any way similar, only that the power will ultimately originate from the people in some way. You don't even have to go ancient for this - there are dozens of different takes on democracy in the modern world.
To emphasize - I'm not arguing against the distinctions or the new ideas that were made during the founding of the US, or that some of these ideas weren't novel. I'm arguing against the school of thought that seemingly arose long after that happened, which tried to categorize the US into a bespoke category of government that included it and excluded other nations, modern and historical - despite them all definitionally being ruled by the people and often even having similar ways of organizing their governments. Let's review the terms.
An autocracy is a state where power descends from a single person, the autocrat. An absolute monarchy is a subset of autocracy, where that person is a monarch, usually with religion tying into their perceived legitimacy. An aristocracy is a state where power descends from an upper elite class. In anarchy, the power descends from no one in particular. Democracy is the class that is opposed to these other forms, it's a system of rule by many where power descends from lots of people. It includes many flavors and forms that result in different types of government - direct, indirect/representative (you are here), as well as many sub-variants that define the exact way of how power moves from the ruled to the rulers or other characteristics.
Therefore, republics are a strict subset of the broad term of democracy. There are lots of republics in the modern world, all of which are democratic states, just as there are constitutional democracies. If you want to distance the US from these other states, you would need to prove why it was not envisioned as a power-from-the-people state and what exactly would place it in a different category that no one else occupies. The US may have been the first, but it's not that unique in the modern landscape of governments.