Praise be to Allah, The Lord of The Worlds ;and may His peace and blessings be upon Prophet Muhammad and upon His Family and companions.
Islamic Sharia forbids calling a person by a name other than that of his real father because Allah, The Almighty, Said (What means): "Call them by (the names of) their fathers: that is juster in the sight of God. But if ye know not their father’s (names, call them) your Brothers in faith, or your maulas." {Al-Ahzab/5}.
It also forbids that a person attributes his fatherhood to anyone besides his real father because the Prophet (PBUH) said: "He who (falsely) attributes his fatherhood to anyone besides his real father, knowing that he is not his father, will be forbidden to enter Jannah." {Related by Bukhari & Muslim}.
Accordingly, it is incumbent on those who falsely attribute the above girl to other than her real father to fear Allah and free themselves from the liability by setting the records straight. They must also stop manipulating the rules of Sharia since a human being is Allah`s, and none has the right to gift him/her to another because one can`t gift that which isn`t his.
In fact, the above persons should have allowed that woman to raise her niece while attributing her fatherhood to her real father. Anyway, the above questioner can only advise the above girl and her real mother with ways that are best and most gracious.
If they don`t listen to him then he is free from liability since Allah Burdens not a soul beyond its capacity.
It is worth pointing out that the aunt`s husband is a non-Mahram (Marriageable) to that girl, and so he isn`t allowed to look at her, and neither he nor she inherit each other since she is eligible to inherit only her real father, and this fact must be made crystal clear to all parties involved. And Allah Knows Best.
