Q. Should an adopted child say Kaddish for their adoptive father or mother?
A. The answer requires an understanding of the status of adoption in Jewish law. Adoption of some kind was known from Biblical times; Moses was brought up by Pharaoh’s daughter, Ruth’s son by Naomi and the sons of Merab (Saul’s daughter) by their aunt. Judaism sees it as a great act of piety and humanity to rear a neighbour’s child (Meg. 13a, Sanh. 19b).
But though poetically the sources say that one who performs this mitzvah is as if they gave the child birth, adoption is not a halachic institution and the birth relationship is not eliminated. Thus the child’s religion is determined by its natural mother, and in theory Kaddish is required for a natural parent if their identity is known. There is no strict duty of saying Kaddish for an adoptive parent, but as a mark of love and respect an adopted child will certainly want to undertake it.