Is it permissible to marry a girl who was adopted and raised by a man, while she still carries her original father's name, and does she take the ruling of a stepdaughter (rabībah)?

If what the guardian does is merely kindness, upbringing, spending, and educating, while the girl retains her lineage to her father, then it is commendable شرعًا (religiously) due to the merit of sponsoring an orphan. There is no impediment to the guardian marrying this girl unless his wife had nursed her a forbidden suckling within the first two years, for then she would be his daughter through suckling. If her marriage is permissible, the consent of her guardian (her father or whoever acts on his behalf) is necessary. This girl is not like a stepdaughter whose marriage is forbidden, for a stepdaughter is the daughter of a man's wife from another man.