Star Trek - What is the relationship between the 'Bread and Circuses' Roman planet and the Romulans?

In Startrek TOS Episode "Bread and Circuses" we see a planet with Roman Governance and gladitorial games.

In the Startrek Canon - we meet the Romulans, a race of people with a Roman governance model.

What is the relationship between these two?

(Only hypothesis I have heard is that the Roman gods were a several thousand year old advanced Alien race that visited all three planets and influenced them to varying degrees).

Answer

There isn't one.

Gene Roddenberry had an interest in ancient Rome, so it was simple coincidence (bolding mine):

Paul Schneider modeled the Romulans on the ancient Romans, naming the species' homeworlds after the mythical founders of Rome, Romulus and Remus. "It was a matter of developing a good Romanesque set of admirable antagonists that were worthy of Kirk," Schneider related. "I came up with the concept of the Romulans which was an extension of the Roman civilization to the point of space travel, and it turned out quite well." (Captains' Logs: The Unauthorized Complete Trek Voyages, p. 34) D.C. Fontana reckoned that Schneider basing the aliens on the pre-existing Roman civilization was the cause for the writer receiving insufficient credit for creating the Romulans. ("Balance of Terror" Starfleet Access, TOS Season 1 Blu-ray) Gene Roddenberry, interested in ancient Rome himself, approved of the initial depiction of the Romulan species. "He loved Paul's having endowed the enemy-Romulans with the militaristic character of the ancient Romans," wrote John D.F. Black and Mary Black. (Star Trek: The Magazine Volume 2, Issue 11, p. 19)

The "extension of the Roman civilization" part can only be interpreted as "this is how their society is modelled", not a literal extension of our Romans, because Romulans were never human.

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The theories you've heard probably stem from mixing up the Greek and Roman gods, as one Greek god, Apollo, does appear in TOS 2x02, Who Mourns for Adonais?. The Romans seem to have adopted Apollo as their own god as well, with the same name, but within the episode he identifies himself as Greek.

However, in-universe we have Hodgkin's Law of Parallel Planetary Development which means we don't need Roman godlike aliens to explain away the similarities.