Is it permissible to adopt a foundling and give him the family surname to spare him future embarrassment?

If the father of a child is unknown and no one claims his lineage, then he is not to be attributed to anyone. Rather, he is to be called a brother in faith, for Allah the Almighty says: ﴿Call them by [the names of] their fathers; it is more just in the sight of Allah. But if you do not know their fathers, then they are your brothers in religion and your protectors.﴾ Attributing him to the lineage of his guardian is strictly forbidden. However, his guardianship, upbringing, and providing for him are among the greatest acts of closeness [to Allah].

If there is no fosterage relationship between the child and his guardians, then he is considered a stranger (non-mahram (definition)), and the rulings of seeking permission, looking [at them], and seclusion must be applied. It is preferable to establish a mahram relationship through fosterage; so, the woman suckles the child, or her sister or female relatives suckle him, so that he becomes a mahram to her, and interaction within the family becomes easier.