Islam is keen that every child is to be named after his or her biological father, whether the child is an adopted child or not. Thus, if a man adopts his wife’s child, in the sense of caring for his well-being and securing an appropriate upbringing for him, the child should retain his biological father’s name and not take his stepfather’s name. This is because Islam commands that we call people by their proper names, not by false names.
The Islamic Religious Council of Singapore states the following: Every child is born in a state of fitrah or natural disposition, that is pure from polytheism. It is his parent who will change that original state of purity. If one has adopted a non-Muslim child, and he has not attained his puberty yet, there is no need for you to ask for his shahadah (Testimony of Faith), since he is still in the state of fitrah. If he has attained his puberty, you should require him to profess the shahadah. It is forbidden to put one’s name after the adopted child’s name since this will deceive him and others into thinking that he is one’s true child. Allah Almighty says: [Proclaim their real parentage. That will be more equitable in the sight of Allah. And if ye know not their fathers, then (they are) your brethren in the faith, and your clients. And there is no sin for you in the mistakes that ye make unintentionally, but what your hearts purpose (that will be a sin for you). Allah is Forgiving, Merciful] (Al-Ahzab 33:5).
If you do not know your adopted child’s father’s name, then we can put “bin Abdullah” at the end of his name, provided that the adopter’s name is not Abdullah or we can just leave out the “bin” part.