If adoption is forbidden in Islam, then how did the Prophet, may Allah bless him and grant him peace, adopt Zayd ibn Harithah and say about him: "He inherits me and I inherit him"?

Adoption (تبني) is forbidden in Islam. It was practiced before but was abrogated by the verse: "Call them by [the names of] their fathers; it is more just in the sight of Allah." The Prophet's adoption of Zayd ibn Harithah occurred before the revelation of the verses prohibiting adoption. Zayd had been captured and sold to Hakim ibn Hizam, who gifted him to his aunt Khadijah, who then gifted him to the Prophet. The Prophet freed him and adopted him. When his father and uncle came to ransom him, the Prophet gave Zayd the choice, and Zayd chose to stay with him. The Prophet then made those present witnesses that Zayd was free, his son, and would inherit from him and be inherited from him. He continued to be called Zayd ibn Muhammad until the revelation of the verse, after which he was called Zayd ibn Harithah.