He married her then he found out that her name was false
It is not one of the essential parts or conditions of marriage that correct names be used; rather it is sufficient to specify the woman to whom marriage is proposed and with whom the marriage contract is done. If there is some error in the name or a false name is given, that does not affect the validity of the marriage.
It says in al-Sharh al-Mumti‘ by Shaykh Ibn ‘Uthaymeen (49, 48, 12), in a discussion on the conditions of marriage:
Specifying the partners: because the marriage contract has to do specifically with them, the husband and wife. It is a serious matter on which rulings on lineage, inheritance and rights depend. Hence it is essential to specify who the partners are. It is not acceptable to say “I give my daughter in marriage to one of your sons” or “I give my daughter in marriage to one of these two men: of “I give my daughter in marriage to a student in the college.” Rather it is essential to be specific. And it is also essential to be specific with regard to the wife, and say (for example): “I give you my daughter in marriage.”
There are several ways of specifying:
(i) By pointing and saying “I give this daughter of mine to you in marriage,” and (the groom) says, “I accept.”
(ii) By naming her, such as saying “I give my daughter Faatimah to you in marriage,” and he has no other daughter by this name.
(iii) By mentioning her with some distinguishing characteristic, such as saying, “My daughter who completed grade ten this year” or “My tall/short/fair/dark/blind daughter” and so on
(iv) By mentioning some real fact, such as saying “I give my daughter to you in marriage,” when he has no other daughter.
So in terms of validity, the marriage is valid.
But what this woman did is something that is haraam and there is no difference of scholarly opinion with regard to that. For a Muslim to attribute himself to someone other than his father is something that the scholars regard as a major sin, because of the stern warnings that are narrated concerning it. This has been discussed previously in the answers to questions no. 638 and 1942.
What she has to do is repent from that action of hers and correct her name. She should fear Allah when speaking and not speak anything but the truth, for speech may lead to Paradise or to Hell. The Muslim has to strive when speaking to seek the truth, justice, honesty and fairness.
And Allah knows best.