Is it permissible to secretly marry the mother-in-law of one's wife – who raised the wife but is not her biological mother – with the intention of alleviating her loneliness and deteriorating psychological state, knowing that the wife is not the biological daughter of the mother-in-law by birth or by rada'a (nursing)?

If the wife was adopted, this does not legally prevent marriage to her unless there is a legal impediment. Although adoption is forbidden in Islamic law, one should exercise caution and consult with those of sound judgment to consider the benefits and harms involved, especially regarding the establishment of lineage and legal accountability. The woman must be attributed to her real father, as Allah Almighty says: "Call them by [the names of] their fathers; that is more just in the sight of Allah." It must also be noted that the condition for polygyny is the ability to deal justly with wives.