Praise be to Allaah.
Before answering this question, it will be useful to shed more light on the issue of adoption and giving a child a name or lineage that is not really his.
In Islam, adoption (i.e., giving your name to a child who is not yours) is haraam, and is considered to be a major sin because of the wrongdoing and corruption that result from it. Allaah says (interpretation of the meaning): "Call them (adopted sons) by (the names of) their fathers, that is more just with Allaah" [al-Ahzaab 33:5].
The Messenger of Allaah (peace and blessings of Allaah be upon him) said: "Whoever claims knowingly to belong to someone other than his father will be denied Paradise." (Reported by al-Bukhaari and Muslim, Fath, no. 3982). He also said: "Whoever claims to belong to someone other than his father, will be cursed by Allaah, the angels and all the people, and Allaah will not accept any deeds or excuses from him on the Day of Resurrection." (Reported by Muslim, may Allaah have mercy on him, in his Saheeh, no. 2433).
In his commentary on this hadeeth, al-Nawawi said: "This is clearly stating the emphatic prohibition of claiming to belong to anyone other than one's real father because this involves ingratitude and a denial of the rights of inheritance, as well as cutting family ties and undutifulness to parents." (al-Nawawi's statement ends). It also involves not carrying out the rules of inheritance, and ignoring the prohibited degrees of marriage to women [i.e. a man may unknowingly marry someone he is not permitted to, such as a sister, when true origins are concealed by not calling a child after his real father - translator's note], etc.
If a man wants to sponsor an orphan, the child must still be named after his father, if his name is known; it is not permissible to attribute this child to himself by giving him his name. If the child's real origins are not known, then he should be given an appropriate name by which he may be known to others, as Allaah says (interpretation of the meaning): "Call them (adopted sons) by (the names of) their fathers, that is more just with Allaah. But if you know not their father's (names, call them) your brothers in faith " [al-Ahzaab 33:5]
In answer to your question, which has to do with a girl - for example - addressing the one who is sponsoring her and bringing her up as "Father" or "Daddy," Shaykh 'Abd al-'Azeez ibn Baaz has issued a fatwa (ruling) stating that this is allowed, because he is acting in the position of a father to her, and we should not be so strict in this matter.
Note: If a man is sponsoring a very young girl (less than two years old) he should try to establish the relationship of mahram by letting her breastfeed from his wife, mother or sister, so that she will become his mahram, otherwise he has to make sure that she does not uncover (take off her hijaab) in front of him when she grows up. The exception to this is a step-daughter (his wife's daughter from a previous marriage); she does not have to wear hijaab in front of her mother's husband even if no relationship is established through breastfeeding (al-radaa').