I think the unit types should be about half the same. The troop tree should have an overhaul if this is done, so there would be some new fun units to distribute to these factions. For example, we could have recruits transformed into a dedicated skirmisher class like a "velite" (also skirmishers should have a major rework regarding javelin damage and overall equipment and stats).
A possible way to divide these:
Northern Empire
- Focus on cav, maybe unique access to cataphracts or other cavalry because the North is traditionalists who think The Empire should be ruled by the nobles in the senate and cav are usually nobles. Dunno.
- They could also have access to mercenary Nord soldiers as shock infantry instead of menavlions to spice things up.
Western Empire
- Focus on infantry, since the leader in lore is a general who is loved by his legionaries. Maybe the west could have some more infantry variants or a modified legionary*.
Southern Empire
- Possible focus on archery or skirmishers (the southern empire lore I think has some greek influence), and maybe the south should have unique access to palatine guards. This is because the faction is lead by an empress and they are royalists, so the palatine guards would be a good fit.
* The staple empire troop, legionaries, should be available for all of these factions, but could have slightly different outfits. The north and south could have mail armor and no access to the "improved armor" perk. The western legionaries could have mail armor as standard, but could be improved to have the heavy plated armor which they have as "improved armor" right now.