Inheritance is not connected to breastfeeding
Question

A woman died, and she has no relatives who could inherit from her, but she has a son through breastfeeding. Is he entitled to a share of the estate?
Praise be to Allah.What makes one entitled to inherit is known and is  clearly defined in Islamic teachings. Some of these factors are something on which the scholars are unanimously agreed, and they are three: marriage, blood ties and manumission.
There are other factors concerning which the scholars differed, namely: alliance, if someone becomes Muslim at a person’s hands, and one who takes in a foundling; and the Muslim treasury (bayt al-mal) inherits the estate of one who has no heir.
Shaykh Salih ibn Fawzan al-Fawzan explained the view regarding these factors in his book at-Tahqiqat al-Mardiyyah fi’l-Mabahith al-Fardiyyah, p. 31-44.
Breastfeeding is not one of these factors that make someone entitled to inherit.
So if a woman died and had no heir, her wealth goes to the Muslim treasury (bayt al-mal), and her foster son is not entitled to inherit from her.
Shaykh Muhammad ibn `Uthaymin (may Allah have mercy on him) was asked:
If a woman dies and she has no heir, and the closest person to her is the one whom she breastfed, whether it is a man or a woman, is that person entitled to inherit her estate, or does it go to the Muslim treasury (bayt al-mal)?
He replied:
The connection established through breastfeeding is not one of the factors that make a person entitled to inherit. So a person’s brother through breastfeeding or father through breastfeeding are not entitled to inherit from him, cannot have guardianship of him, are not entitled to maintenance from him, or any of the rights of blood kinship. But undoubtedly he has some rights that should be honoured; however, he has no right to inherit. That is because the factors that make a person entitled to inherit are three: blood ties, marriage and manumission; breastfeeding is not one of them.
Based on that, the estate of the woman mentioned goes to the Muslim treasury (bayt al-mal), and this foster son is not entitled to it.
End quote from Fatawa `Ulama’ al-Balad al-Haram, p. 334.
And Allah knows best.