I assume the time period they are tying this claim to is the late republic, as that is when the game takes place. I was under the impression that any male citizen could vote regardless of serving in the military.
Haven't played the game, but what this is might be referring to is that if you were not a citizen of the Roman Empire but lived in its territory, you would be a peregrini and could serve in the auxilia. After serving your full term, 25 years under Augustus, you and your family and your descendants were granted citizenship, which brought a number of privileges and rights, including the ability to vote.