The Roman Republic. They had separate judicial, legislative and executive branches.
The roman republic did not resemble our republic. First, it treated women like dirt (every man had a mistress for pleasure - a wife was used only for procreation to give the man an heir). The senate was all patricians. They did not have full representation, and these patricians were not elected -they inherited their positions or were appointed. In addition, they had no constitution, and did not hold to principles of basic rights for all men (life, liberty, property). In addition, there were no checks and balances and the republic deteriorated into dictatorship in 44 BC. Then, Caesar became arbitary ruler over everybody and everything and no one had any rights that Caeser did not give them. There are other differences, but our founders studied Rome very carefully and did not want to make the same mistakes as Rome did (or Greece).
I might add that, ironically, homosexual marriage was not allowed even in depraved rome. Modern France, Netherlands and the State of Massachusetts should be proud - they are plunged to moral depths not before realized in recorded history!
It resembled it, but it wasn't the same. Every political system in the world is different. You asked for examples of other republics/democracies with certain characteristics, and I provided them. If you want to argue that the US system is unique in human history, I have to agree. However, the same argument could be made about the British system, or the Japanese system, or the Venetian Republic.