The Romulans were influenced by Rome, but also by other cultures, perhaps including Imperial Japan.
But what is most important in making a species believable, is making their culture rational and pragmatic. So, the Romulans are not a like-for-like model of the Roman Empire in space. Rather, we can perhaps infer, they have arrived at a similar culture through practical necessity.
Rome's system arose because it's rulers desired conquest, but had no justification for calling Egypt or Greece a part of Rome - they were foreign lands, and everybody knew it - and there was no Eastern concept of 'universal empire' in which nations became irrelevent, because Rome was rational and not given to idealistic flights, so instead the new territories became provinces, overseen by the senate, with regions governed by Proconsuls, etc. Wars were justified in terms of defence, often, as is necessary in an imperial democracy. We can perhaps infer that when the early Romulans entered space, starting as little more than thugs, they needed an imperial system by which a senate that only governed Romulus could govern other species - so their system of an imperial military, and proconsuls, etc, arose.