It has to be noted that Republic (Res publica / Public affair), and Democracy (Δημοκρατία, Δῆμος + Κράτος / People Power) are not the same thing.
A republic can be an oligarchy (or anything else) and still cater for public affairs, but a democracy is a system that is controlled and exercised by the people themselves. In fact the ancient Greek word equivalent to Republic is Πολιτεία (related to Πόλις / City and closely related to the meaning of Civitas as in the rights of a citizen). As such, although there may have been discussion on the similarities and differences of one to the other, these were never regarded to be the same concept.
Of the ancients, Plato (Plato's Republic) and Aristotle (Politics and Constitution of the Athenians) are the two main important Greek figures examining these issues, Plato on the ideal republic, and Aristotle on a cross comparison of all types of goverment. From Romans, Cicero is the one that translated Πολιτεία to Res Publica (in De re publica) also examined the differences of the Roman state to the concepts discussed by Plato and Aristotle.
The meaning of republic as it is today, comes from the Rennaisance Italian states of Venice and Genoa where although they based their states on the Roman Res Publica, the meaning gradually shifted to signify a non-oligarchical/monarchical state (Republica leading to today's Republic, a type of 'representative democracy' of sorts).
In case it's not made clear, there was no simultaineity
of democracy in Athens and Rome, hence there was no simultaneity of democracy observed in antiquity. There was a coincidental removal
of a king in Rome and a tyrant in Athens by a difference of 1 year
but these were different events that ended up with different outcomes, and the abolishment of monarchy does not constitute democracy, there isn't a one person one vote system in place as Paul Cartledge eloquently describes in https://classicsforall.org.uk/reading-room/ad-familiares/democracy-ancient-vs-modern. The impression that both of these were democracies is based on an understanding of Res Publica as a modern Republic (and even a modern Republic is not the same as what the Athenian democracy was), something which is false (e.g. even the Roman Empire was being called Res Publica and certainly having an emperor does not qualify as having a democracy).
If there are similarities to look at these would be between Sparta and Rome, e.g. 2 kings and 2 consuls, gerousia and a senate, ephors and the magistrates, but this goes beyond the scope of the question.