Legally, they're considered waste products. But because there is a value attached to them, we do have very rigorous consenting in the donation, and we use the mother as the person who is giving permission for the cord blood, which legally would belong to the infant.
Interesting legal issues for cord blood units are raised if they are banked past the year of consent, so past the age of 18 or 19, which, to the best of my knowledge, nobody has tried to sort out legally.