I get the impression "Umbra" simply means "shadow"- perhaps in Aldmeris, perhaps in something else (if we're going real-world wise, Latin, obviously).
Keep in mind the creation of Umbra the sword/ Umbra the being: Clavicus Vile, in typical fashion, wanted a plaything that would send him souls. But when he tried to get a witch to create the sword for him, a piece of his soul was necessary to keep the weapon stable. This little piece somehow gained sentience and started calling itself "Umbra", functioning as a magical weapon with a life of its own, possessing its owners.
So perhaps the piece of Vile's soul just called itself "Shadow"- a reflection of power, even it if it wasn't as powerful as its source. Think about Peter Pan and his escaped shadow, which was less than enthusiastic about being sewn back onto its owner's feet. Similarly, Umbra the Shadow is in no rush to reunite with Vile.
i.e., no connection. Just a word.