Also, there's a bit of trickery here in terms of the Empire factions - just because factions have the same name doesn't mean they are the same faction. There can be more than one Empire Party, for example, even in the same system, and they do not share states because they are actually unrelated factions and reputations with one don't effect the other.
Yes, there are multiple minor factions with the same names but different reputation.
Im not exactly sure if a faction that belongs to the big Three counts as I minor faction.
AFAIK the only thing the major factions are are a color on the map and some parent reputation pool that is filled by doing stuff for minor factions and from which a base reputation is inherited from minor factions that you didn't encounter yet.