If the man has children who are poor, it is not permissible for him to give the zakaah of his wealth to them, because it is obligatory for him, according to sharee‘ah, to spend on them and make them independent so that they do not have to ask others for what they need. There are some exceptions to this ruling; for more information see the answer to question no. 105789 and 130207.
This applies to one’s own children and grandchildren. With regard to children through breastfeeding, it is permissible to give them zakaah, because one is not obliged to spend on them.
Ibn Qudaamah (may Allah have mercy on him) said: The scholars of the ummah are unanimously agreed that relatives through breastfeeding are mahrams. With regard to the other rulings connected to kinship through blood, such as spending on maintenance, manumission, rejecting testimony and so on, they have nothing to do with the relationship through breastfeeding, because blood ties are stronger than it, so an analogy cannot be drawn in all rulings; rather the resemblance is only in the matters referred to by the shar‘i text. End quote from al-Mughni.
He also said (8/169): All of the scholars from whom we learned are unanimously agreed that a person is obliged to spend on those of his children who have no wealth. With regard to spending being obligatory, three conditions must apply… the third of which is that the one who spends should be an heir. End quote.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said:
Breastfeeding does not make any of the rulings to do with kinship applicable except those that have to do with marriage only; breastfeeding creates the relationship of mahram, makes it permissible to look (at the person) and haraam to marry that person. But it does not make the people concerned heirs of one another or mean that one is obliged to spend on the other or to pay the costs of diyah or to uphold the ties of kinship, as in the case of blood relatives. Most of the rulings of blood ties do not apply in the case of breastfeeding. End quote from Noor ‘ala al-Darb. This has also been explained in detail in the answer to question no 40005.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked:
Can zakaah be given to one’s mother through breastfeeding and sister through breastfeeding?
He replied: Yes, zakaah may be given to one’s mother through breastfeeding and sister through breastfeeding, if they are entitled to zakaah. That is because one is not obliged to spend on the mother through breastfeeding and the sister through breastfeeding; so zakaah may be given to them on condition that it be proven that they are entitled to it. End quote from Majmoo‘ al-Fataawa, 18/417
And Allah knows best.