Is it permissible, Islamically, for a man to attribute to himself a child born to him from an illegitimate relationship, and is he obliged to marry the child's mother, who is still practicing prostitution and abusing drugs, knowing that the child needs to prove his lineage to be enrolled in school?

We praise Allah for your repentance from this sin, and this is from His immense favor upon His servants, as He accepts the repentance of those who err and forgives their sins, in fulfillment of His saying, exalted be He: "Say, 'O My servants who have transgressed against themselves [by sinning], do not despair of the mercy of Allah. Indeed, Allah forgives all sins. Indeed, it is He who is the Forgiving, the Merciful.'" And His saying: "But indeed, I am the Forgiver of whoever repents, believes, and does righteousness, and then is rightly guided."

As for the child resulting from the illicit relationship, the majority opinion is that he is not your legitimate son, and it is not permissible for him to be attributed to you or for you to inherit from him. Rather, he is attributed to his mother.

You may refer to the fatwas concerning marriage to an adulteress for more information.