There are cases where there is no official adoption sources, none the less the family accepted full moral, legal and physical responsibility for the child. I have several in my database.
To keep the "adopting" family in the loop, I have the children list with the natural parents and with the "adopting" parents where the relationship indicates the children were adopted. These "adopting" parents have significant influence on the child and are very important to the total family.
While there is a legal adoption, there is a common law adoption. (I don't know if this concept appears in any legal system but the phrase was the best way I could think of to describe the situation.)