According to Clavicus Vile, Umbra contains the soul of a person named Umbra. Likewise, the Staff of Hasedoki supposedly contains the soul of Hasedoki, even after all this time.
I believe the essential power of the artifact is sustained by the original soul, and subsequent, often lesser souls are used to recharge its abilities.
Umbra is still in the sword even when the charge is Zero, but it hungers for more souls to continue its vampiric nature of soul eating.