It isn't a coincidence, but not in the way you think. Rome had no contact with Athens in 510, and they probably only knew about them indirectly from the Etruscans. Furthermore, the Roman republic at its inception bears little resemblance to even the more conservative democracy set up by Cleisthenes, before the radical democrats came to power. There's pretty much no reason to suspect that the Athenians were some sort of model

The reason that 510/509 appears is simply due to the Roman tendency to compare themselves with Greeks (which the Greeks did too sometimes--just look at Plutarch). This appears very frequently in early Roman history, since it's essentially prehistory. According to Livy and others the Romans expelled the Tarquins in 510 and founded the rule of the senatorial magistrates (what we call the Republic) in 509. Thing is, in Greece, that's the exact same year, since the new year was calculated on the summer solstice. This means that the Athenian Revolt and the expulsion of the Tarquins followed by the foundation of the Republic took place in the exact same year according to the Greek calendar. This makes it suspect at best, especially since dares so early in Roman history are, unlike Greek history of a comparable period, more or less non-existent