Is what the young man did—sponsoring a child of unknown parentage, registering him in his official documents under his name or the name of one of his wives, with the intention of facilitating the child's affairs and obtaining a visa for him, along with his intention for one of his wives to nurse the child so that he becomes a legitimate son to them—considered a sin? And what is the solution if it is?

Adoption is forbidden and is one of the practices of Jahiliyyah (pre-Islamic ignorance) that Islam abolished. Merely attributing the child formally—on official papers only—is not sinful if there is a legitimate interest for the child in doing so, provided the necessary precautions are taken. This child is not attributed to you. If your wife breastfeeds him within the two years (of nursing), your wife becomes his mother through suckling, and you become his father through suckling.