All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
We have already made it clear that this boy is a stranger to this man; i.e., has no family relationship with him.
So, this boy should be treated like any other stranger to the family. He is not to be considered nor treated as a member of the man's family. But, if this man wants to spend on him, to financially support him, to see that he is educated and raised in a Muslim environment, he may do so as a good deed for Allah's sake, just as he may do for any other person. However, he cannot be considered as a member of the family or as an heir. It may even be a good deed which will be rewarded, Allah willing. This is so, because if this boy is educated according to the Islamic Sharee’ah and with a Muslim woman, he may become a good pious man. If, on the other hand, he is left with a non-Muslim woman in this country of disbelief, he will most probably become a non-believer.
So, it is lawful for you, and you are even recommended to look after this child and to educate him as to prevent him from going astray in this bad environment.
However, you should screen your wife and daughters from him as soon as he reaches the age of maturity.
For more benefit that the child is traced back to the fornicator (or adulterer) if he requests that the child be traced to him, please refer to Fataawa 97254, 83680, 168603 and 84788.
Allah knows best.