the Fatwa Department Research Committee - chaired by Sheikh `Abd al-Wahhâb al-Turayrî
This depends on the reason for adopting the child. Ideally, the child should be raised by his or her own parents, if they are alive. If they need financial assistance, it is better to give that assistance to the parents instead of separating their children from them.
However, in special cases where the parents are abusive, extremely ill, or otherwise unfit to raise the children, then it can be a good deed to take care of that person's children for them. This should only be done if there is:
(1) a court order for the transfer of custody in a case of abuse
(2) the parents expressly and truly wish it, like might be the case due to extreme illness or some other reasons.
The Islamic rulings regarding adopted children must be strictly observed. In Islam, the child remains the child of its biological parents. The one's who adopt the child will not assume the identity of the biological parents.
For instance, the attribution of a child to the one who is bringing the child up is unlawful.
Allah says: "Call them by the names of their fathers. That is more just in the sight of Allah, but if you know not their father’s names, (then they are) your brothers in faith, or your friends but there is no blame on you if you make a mistake therein: (what counts is) the intention of your hearts."
Hence, it is lawful to bring up children in any house but their attribution should always be to their true, biological fathers.
There is no harm in the child calling his or her guardians "Mum" and "Dad" due to affection. This is merely a matter of custom. This is why Allah says: "…there is no blame on you if you make a mistake therein: (what counts is) the intention of your hearts."
Likewise, adopted children will not inherit from their guardians who adopted them, but only from their true biological relatives. These children may marry from their adoptive “father's” or “mother's” families, since they are not truly the children of their adoptive caretakers.
And Allah knows best.
Source: Islam Today