Despite the loading screens noxiously spamming of text from the trials of saint Allesia, Mankar Camoran wearing the Amulet was done intentionally. The reason he or anybody can wear the amulet has little to do with blood relations, but rather with their actions (taken in their mythical context).
So, Mankar could wear it because fate hung in the balance, and he was possibly destined to become the Emperor of a new age?
I know in lore, different dynasties like the Reman and Septim ones have recovered and claimed the amulet as their own when they were founded.
And the Elder Scrolls read by the Moth Priests record fate as something fluid with multiple outcomes. So Mankar and Martin are both potential Emperors.