It's almost certainly a coincidence. For one, accounts of the two revolutions don't cite any kind of ideological motivation and don't reference inspiration from outside sources. Two, both mainly expanded the power of existing institutions (Rome's senate and comitia centuriata; Athens' boule and ekklesia) rather than invent totally new ones.

The Roman Senate (supposedly) overthrew Tarquinius Superbus for the excesses of the royal family, and not out of any kind of radical ideological sentiment. The Senate of leading citizens, the various priesthoods and magistracies dominated by the Patrician families, and the comitia centuriata assembly which grew out of the Roman citizen-militia already had defined political roles before 509 BC and the end of the monarchy just meant creating new offices to fill the king's role. To achieve this they created the office of consul to take on the king's military and administrative powers and the rex sacrorum priesthood to take on his religious duties. Consuls were very much like their Greek equivalent, the archon, and the idea of a dual consulship may have been original, or could have been inspired by either the Spartan dual kingship or the Carthaginian sufetes (who also served in pairs for one-year terms, but did not have military authority or act as generals). The consuls were selected from the ranks of the Senatorial aristocracy and put up to election by the comitia centuriata- in this regard, the Republic was founded more as an aristocratic coup against a weak monarch than as an ideological victory for 'republicanism' as a strain of political thought. The changes in power structure were relatively modest compared to the changes that would come out of the Conflict of the Orders and the Social Wars.

Likewise, in Athens the key organs of democratic power (the ekklesia or popular assembly, the heliaia or popular juries, and the boule or elected executive council) date back at least to the time of Solon. Meanwhile, the rule of the tyrants was, strictly speaking, unconstitutional to begin with. Cleisthenes' challenge was not replacing a legal monarch, but to appease the Athenian masses so as to avoid another illegal dictatorship. Athens already had tribes- Cleisthenes redrew them so that they were based on geography rather than aristocratic clans. He also moved powers previously held by the aristocratic Areopagus council (similar in many ways to the Roman Senate) over to the more democratic ekklesia and heliaia, opened many offices to election by lot, and enlarged the citizen body by enrolling all of the Attic communities as Athenian citizens. These changes were more radical than the ones in contemporary Rome, but still worked through shifting the balance of power between existing institutions rather than on creating new ones in service of a comprehensive ideological program.