After entering last dungeon of this quest (with 3 bosses) i sow that my companion Lidia wears staff but not shild. And wile we were fighting she used staff but not bow as usual. And after ending this quest and exiting dungeon she was wearing shield. I think that staff is hiden item. Your apinion?

They always change. I don't know why exactly they change from Staff-to Shield for... Maybe she doesn't have a sword?

Lydia always uses the staves I give her in preference to the bow - as long as they have sufficient charge(s).

When weapons are sheathed she lugs the shield.

pretty sure she picked that staff up. I emember charging it back up for her , but she didn't use it again after that quest so I took it and stored it in a chest. Keep that amulet. Its a beauty.