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, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
Your son must repent to Allaah The Almighty from the sin of Zina (fornication/adultery). Repentance has certain conditions that we have clarified in Fatwa 86527. It is also incumbent upon him to guard himself against every means that may lead him to commit Zina again such as being in seclusion with a non-Mahram woman, prohibited mixing among men and women and the like. If he has not married yet, we advise and urge him to marry to protect himself from Zina and remain chaste and he should be helped in this regard if he needs help. Muslim jurists are of the opinion that it is obligatory to help the poor son to marry if he needs marriage. Ibn Qudaamah (from the Hanbali school of jurisprudence) said: "Our companions said: The father must keep his son chaste if he is obliged to spend on him and if he (the son) needs help to remain chaste (by getting married)."
If that woman is married, then that girl will be attributed to her husband because he is the owner of the bed. The DNA analysis that you have conducted and the resemblance between her and your son do not count. For more benefit, kindly refer to Fataawa 122944 and 84301.
Based on that, her father (the husband) will be responsible for spending on her and taking care of her. If he carried out what he threatened to do, i.e. using her in prostitution or in anything which Allaah The Almighty has prohibited, surely he will get his punishment from Allaah The Almighty. Ibn ‘Umar said: I heard the Messenger of Allaah say, “Each one of you is a guardian and is responsible for the people under his guardianship, …..and a man is a guardian in his house and is responsible for the people under his guardianship." [Al-Bukhari and Muslim]
As for your son, he has no relationship with that girl at all and it is not obligatory upon him to spend on her. So, you should not become attached to her for she is an unrighteous deed (of your son). It is not permissible for that woman (her mother) or others to exploit and blackmail him to take money from him. In this case, he can threaten them to refer the matter to the concerned authorities if he expects that this will stop them doing so.
Allaah Knows best.