Is nursing that results from injecting the wife with hormones to produce milk considered legitimate nursing in Islamic law, and does the child thereby become one of her mahram?

The majority of scholars hold that the milk of a woman that results in fosterage does not necessarily have to be due to pregnancy or sexual intercourse. Rather, if milk is produced due to consuming something, and she then breastfeeds a child with it, this suckling child becomes her child through fosterage, provided there are five feedings and it occurs within two years of age or less. This is the madhhab of Malik, Shafi'i, and Abu Hanifa.

Accordingly, if a woman takes beneficial medications or foods that cause her to produce milk, and she breastfeeds a child five times with it while the child is two years old or less, then the child becomes her son/daughter through fosterage. This establishes the prohibition of marriage, the permissibility of seclusion and looking, and mahramiyyah for travel. However, it does not establish lineage, the obligation of financial support, or inheritance.

As for the husband of the breastfeeding woman, he does not become a father through fosterage because the milk did not originate from him. Therefore, his children from a wife other than the breastfeeding wife are permissible for the foster-daughter, and his daughters from a wife other than the breastfeeding wife are permissible for the foster-son, except concerning practical legal rulings such as looking and seclusion, as he is the husband of the foster child's mother.

The word "adoption" (التبني) mentioned in the question is not in its forbidden legal sense; rather, it refers to fostering an orphan. It is preferable not to use it due to the ambiguity it creates.