Yeah I’ve been wondering about the long axe. I think the fatigue recovery is granted by the polearm perk but the axe perk probably gives you a bonus to the split shield.
If you take a look at a weapon’s tooltip, you’ll see its classification. The Longaxe, for example, says ‘Axe, Two-Handed’, meaning it’s governed by the Axe Mastery perk. Works like this for every weapon in the game.