We are not Azem, especially not when Azem exists in the story as a distinct (if unseen) character who is talked about separately. We do not have their powers or their memories, or even their entire soul. Emet and Hyth are long-term friends with Azem but have only just met us.
Just because we look like Azem and remind people of Azem does not make us Azem. We are the inheritor of their legacy but a separate person.
It doesn't serve as a "universal noun for the player character" because it refers to someone else.