but it doesn't count to weapon skill

in the game, you're either chopping wood, or you're breaking skulls.

But in reality, there may be a great skill in using an axe as a weapon and even greater skill in using it as a tool, but you're body is still developing the abiltity to wield said tool/weapon.

So regardless what you're hitting, just the fact you are wielding the same thing that is either a weapon or a tool should improve something that crosses both skill sets.