The Quran is clear in allowing adoption. The only stipulation is that we are advised to retain the child's true identity by what we name them as this is seen as more just and honest:

...Nor did He make your adopted children to be your sons. Such is what you claim with your mouths, but God speaks the truth, and He guides to the path.
Name them by retaining their fathers names. That is more just with God. But if you do not know their fathers, then, name them as your brethren in the obligation and your friends. There is no sin upon you if you make a mistake in this respect; but you will be responsible for what your hearts deliberately intend. God is Forgiver, Merciful. [33:4-5]

However, that is not to say they should be treated any differently in terms of affection and care, on the contrary, they are to be considered as one of our own:

...and they ask you regarding the orphans, say: “To fix their situation is best, and if you are to care for them, then they are your fellows.”... [2:220]

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