Is it permissible, in Sharia, to treat a young man raised in the home as a brother, or does he remain a stranger to the questioner and her children, and how should he be dealt with?

If the reality is as stated, then this boy is a stranger to you and to your daughter. He is not a brother to your husband nor to his brothers. Therefore, it is not permissible for you to be alone with him or to not observe hijab (definition) in front of him. This is unless your husband's mother breastfed him five full feedings while he was under two years old. In that case, he becomes her son through suckling, a brother to your husband, and an uncle to your daughter through suckling. However, he still remains a stranger to you. And one must beware of adoption, for it is forbidden.