Quest items that are equippable are still flagged as impossible to drop, sell, or store. Even after a quest item loses relevance in that way, it remains flagged as such, so even once the Tagos' Axe quest is resolved one way or another, it is marked as a quest item.
Fortunately, these quest items do NOT count towards your inventory limit. The only annoyance they pose is having them in your inventory list.
