It is legally impermissible in Islamic law to give the adopted child the full name of the foster family which would be reflected in the legal rights that the adopted child will share with the adoptive family. God says in the Quran “Call them by [the names of] their fathers; it is more just in the sight of Allah . But if you do not know their fathers - then they are [still] your brothers in religion and those entrusted to you.” 33:5

It is also well known that the noble companion Zayd ibn Haritha was previously known as Zayd ibn Muhammad as he was adopted by Prophet Muhammad who treated him like his own child and attributed his name to him. After God prohibited attributing the name of the foster father to the adopted child, the Prophet returned Zayd’s name back to his biological father and was known for it.

It is legally permissible to give the adopted child only the family name of his foster family which guarantees loyalty of the adopted child to his new home without claiming biological ties with the adopted child which is prohibited in Islamic law.

Adding the family name of the foster family would be more like the attachment of loyalty which was popular among the Arab tribes. There is no legal impediments to such act as the prohibited adoption would be to fully give the child the name of the foster father as if the child is his own which has legal repercussions in terms of inheritance, lineage, rules of modesty with foster sisters..etc.

As for the verse in which God says “Call them by [the names of] their fathers; it is more just in the sight of Allah . But if you do not know their fathers - then they are [still] your brothers in religion and those entrusted to you.” 33:5, Ibn Kathir in his Quranic exegesis stated that calling the adopted child “son” out of love and kindness is not prohibited and he used the prophetic tradition which was reported by Muslim in which the Prophet called his servant “Anas” as “son”.

Therefore the mere fact of attaching the family name of the foster family to the adopted child, which does not have any legal repercussions, is not prohibited in Islam. In Islam attributing the family name to a person who does not belong to the family biologically can be done for a number of reasons:
- Emancipation: when a slave is freed, he usually takes the family name of his emancipator such as Abu al Bukhturi al Tai who got his last name from the tribe of his emancipator.
- Pledge of allegiance: this pledge is usually formed among tribes for mutual support and defense against attacks. So it is common for the people of the two allied tribes to take the name of the allied tribe as their last name.
- Pledge of companionship due to business or knowledge: it was common if a man stayed long enough in the company of another for the sake of learning and gaining knowledge or if they had commercial dealings to take his family name.
- Pledge of religion: if a man was guided to Islam through another man, it is permissible to take the latter’s family name.

Ibn al Salah, the renowned scholar, stated that the attribution of a man to a tribe which is not his own was common. For example there were people who did not take their father’s name but actually took their mother’s name such as Sharhabeel ibn Hasanah and others took their grand mother’s name such as Ya’li ibn Muniah and others took their grand father’s name such as Abu ‘Ubaydah ibn al Jarrah and others were attributed to men who were not their fathers such as al Miqdad ibn al Aswad who was raised in the house of Al Aswad ibn ‘Abd Yaghuth.

So these are examples which prove that it was common for a man to take other people’s names as long as it does not entail legal repercussions which deem such act impermissible. Same goes for the adopted child when he takes the family name of his adoptive family. Taking the family name especially for a child with an unknown lineage is crucially important for his psychological wellbeing and his healthy upbringing.

Therefore, in your case the illegal relationship which you had with the first man is considered adultery and lineage is not proven through adultery. In other words your daughter is not legally considered to be his legal daughter and therefore it is permissible to give your daughter the family name of your husband but not his first and second name.