"Alb" comes from Latin "Alba". Etymologically speaking, I'm guessing that's where we got the root for the term "albino".
Correct! The word "album" (“blank white writing tablet”) has also that Latin root! All the other major Romance languages (apart from Sardinian and Romansch) have adopted a word borrowed into the Vulgar Latin from Old Germanic as their designation for "white" (blanc/bianco/blanco etc)
"Albină" is from a different Vulgar Latin word, "alvina", which in turn was derived from the Classical Latin "alveus". It originally meant beehive but in the variety of Latin that evolved to become the Romanian language it would change its definition to bee.