All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
This child saying to you “O my father” as a way of honoring you is permissible; however, it is not permissible to attribute him to you because adoption and tracing back someone to other than his father is forbidden as we clarified in Fataawa 84422 and 82371.
The fact that this child has no contact with his father or that his father does not spend on him and so forth are not a sound reason for attributing him to other than his father. Besides, we did not understand what you meant by mentioning the story of Zayd Ibn Haarithah . His story is evidence that proves that what we mentioned about the impermissibility of tracing back a child to other than his father is correct. Allaah Says (what means): {Call them by [the names of] their fathers; it is more just in the sight of Allaah.} [Quran 33:5] So, he was known as Zayd Ibn Haarithah after he had been known as Zayd Ibn Muhammad by virtue of his adoption. Indeed, this did not have any negative effect on him.
As regards the future of those children and preparing a righteous environment that will have a positive effect on the relationship between them, then it is your role and the role of their mother by being wise in dealing with them and clarifying for them that their relationship towards each other is that they are brothers.
Allaah Knows best.