I would argue that the biggest similarity between the Roman and Athenian political systems lies in the historical context which underlines the evolution of Athenian democracy and the Roman Republic. To a certain degree, it does appear that both systems were designed to try to address similar challenges and answer a similar question, although their respective answers were radically different from one another.
It should be noted that Athens was not always a democracy. In fact, Athens had previously been aristocratic and was one of a number of Greek polises that fell under the control of Tyrants—individuals who were able to seize control of a given Polis. This history is important, because part of what lent Athenian democracy its radical nature was this history, because it was not always democratic and did fall under the spell of strongmen (most notably Pisistratus). Ideas like Ostricism, or the selections by lot—Athenian democracy was designed in such a way that whenever possible, power would be diluted to such a degree that future Tyrants and Oligarchs would never be given the chance to emerge. Democracy evolved in response to real questions and challenges.
Likewise, we see in Roman history a similar dynamic, with the Republic being preceded by a quasi-legendary era of kings, and as with Athens, there are certain elements of the Roman Republic which seem designed to minimize the centralization of power. Most notably, there was the selection of two consuls, each for a term of only one year, and each of whom had the ability to veto one another's actions.
Of course, the political systems themselves were quite different. Still, it does seem to me that there's an argument to be made that the two political systems had to evolve to answer similar questions and similar concerns, even if they came to answers very different from one another.