If I don't have access to their mental state, they are not me. I think I am like an instance of an object, and my identity is related to my path through space and time. To be me, they'd have to be standing where I stand, and have stood where I stood.
A 100% copy of the mona lisa is still not the mona lisa--it lacks the history.
You don't always get what you want. When mona lisa gets destroyed 100% copy becomes just as good with it's own history of being perfect copy of something that had long history.
How much "like" you would a copy have to be before it caused you ontological distress?