There were indeed weaponized quarterstaves back in the medieval ages. And while there was no recorded massive use for quarterstaves during wartime, other evidence states that they were used for duels and/or training exercises. We don't have enough historical evidence that states otherwise. So combining the man at arms lower class leather armor look with this weapon makes sense from a historical standpoint.
http://jwma.ejmas.com/articles/2001/...herty_0501.htm
http://www.medieval-life-and-times.i...arterstaff.htm
The "pole axe" in For Honor is more akin to a Halberd than it is a medieval Pollaxe IMO. Even though the term "pole axe" can be loosely used to fit a halberd description and vice versa, I see axe blades in game such as Vogues, Bardiches, Swiss Halberds, German Halberds, etc with various rear fittings such as flukes, spikes, counterweights, etc. No one is really wrong here by definition.