Hi Ryan,
I'm happy to hear that you are enjoying Courier.
Great question. This is an incorrect rule and should be the following:
Friendship Band - +1 Fame or Infamy when Fame or Infamy is gained for your current companion's Faction.
Influence was a system I had planned where you could increase the bond between you and your companions. This would open new perks and abilities to choose from but was changed at the last minute to just be XP gains that coordinate with the player character's XP gains.