Is the adopted mother's failure to breastfeed the child she adopted a reason for her not to be considered his mahram, and for it to be obligatory for her not to appear before him? And what are the rulings related to this situation?

Adoption was permissible at the beginning of Islam, where the adopted child was attributed to their adopter. Then, it was forbidden, and it was commanded that every person be attributed to their father, based on the Almighty's saying: ﴿Nor has He made your adopted sons your [real] sons. That is [merely] your saying with your mouths, but Allah says the truth, and He guides to the [right] way. Call them by [the names of] their fathers; that is more just in the sight of Allah.﴾

Accordingly, families who have adopted children must correct their lineage in official documents and return them to their biological fathers if known.

Based on this, there is no kinship relation between the adopted child and the adopting family. Therefore, they are strangers to each other, and it is not permissible to interact with them as one would with a mahram family, unless there is a mahram-forming suckling.

It is incumbent upon someone who has been adopted by a person to show dutifulness to that family and care for their well-being, due to their favor upon him.