People think that after adoption of a child it becomes a child like other blood childs but it is a misconception because Islam differentiates both.
Islam allows adoption of children but it doesn’t allow the adopted father to declare himself as the father in official terms, in certificates and by law the son belongs to the father from whom he was born.
So if the father dies, who gave his child to someone else for adoption, his son will receive his share from the inheritance (wirasat) even though he never lived with him.
Similarly if the father who adopted the son dies the son will not inherit anything from his property, except those things which the father gifted to that child when he was alive.
The same question was asked to Darul Iftah of Jamia Binoria to which the Mufti said that the adopted son will not inherit anything from the parents which adopted him.