Shogun 2 Complete Edition should have all unit types unlocked from the start in multiplayer.

[Akfiz]

From store page:

​Multiplayer

Create a persistent, upgradable general in Avatar Conquest Mode. Win online battles, earn XP and unlock better armour, new unit types, and permanent upgrades for your troops. Jump into Co-op or Competitive Campaigns to play through the entire Shogun 2 campaign by conquering Japan together or competing for supremacy. Up to 8 players can join hosted lobbies for massive, chaotic skirmishes in Custom Battles.

This was one of the few annoying features of Shogun Total War. You should have all unit types unlocked from the start in multiplayer like in other Total War games.