Praise be to Allah, and may peace and blessings be upon the Messenger of Allah. To proceed:
It is established in the two Sahihs that people will be attributed to their fathers on the Day of Resurrection. This includes the statement of the Prophet, peace and blessings of Allah be upon him, concerning a treacherous person: “This is the treachery of so-and-so, son of so-and-so.” Therefore, the scholars have stated that the basic principle is that a person will be called on the Day of Resurrection by the name and lineage by which he was known in this world.
As for a child born out of zina, the matter is not the same in every case:
If his mother was married, and he was born within the marriage, and the husband did not deny paternity through li'an, then according to Islamic law he is attributed to the husband, based on the Prophet's statement, peace and blessings of Allah be upon him: “The child belongs to the marital bed, and the adulterer gets the stone.”
If his mother was unmarried, and then the man who committed zina acknowledged the child as his and the child became attributed to him and known by that attribution, a group of scholars held that such attribution is valid in this world. Based on this view, it appears that he would be called by this attribution on the Day of Resurrection, because the purpose is identification and distinction.
If the child's father is not known, then it appears that he will be attributed to his mother, since this is the lineage by which he was known in this world.
The scholars have also stated that the reports claiming that people will be called on the Day of Resurrection by their mothers do not have an authentic hadith establishing this. The basic principle is that a person will be called by the identity and lineage by which he was known in this world, whether through his father or his mother.
Accordingly, it is not correct to state categorically that every child born out of zina will be attributed to his mother on the Day of Resurrection, nor that every such child will be attributed to the man who committed zina. Rather, the matter differs according to whether the lineage was legally established and by which lineage the person was known in this world.
And Allah, the Most High, knows best.