What is the religious position regarding an adopted person who does not know their lineage and is unable to ascertain it, after learning that adoption is forbidden in Islam?

You bear no sin for being attributed to someone other than your father due to ignorance. However, you must correct your lineage once you gain knowledge, as Allah the Almighty says: "Call them by [the names of] their fathers; it is more just in the sight of Allah." If knowing the father is impossible, then you should be attributed to a general name that does not suggest affiliation with a specific family. As for the woman who raised you, she is a stranger to you unless she breastfed you or breastfed someone who is unlawful for you [to marry]. It is permissible to be kind to her without seclusion. Her sister or another woman can breastfeed you to establish mahramiyyah, if possible. This is the view of Aisha, Ibn Taymiyyah, and Al-Shawkani for cases of necessity, although the majority prefer otherwise. The rulings of lineage and inheritance do not apply to you under any circumstances.