What is the validity of my brother's claim that he is marrying a woman in a valid marriage according to the Malikis (invalid according to the Hanafis) without the knowledge of her family or witnesses, merely with the knowledge of neighbors, the doctor, and those around them?

Marriage has pillars without which it cannot be contracted: the guardian (wali), witnesses, and the formula (sigha) indicating the contract. What was mentioned about your brother with the woman presents two situations:

1. The marriage contract did not occur at all, and the intercourse is clear fornication (zina). 2. The contract occurred without a guardian or witnesses, and it is invalid.

What your brother attributed to the Hanafis and Malikis is incorrect. The Hanafis do not stipulate the guardian if the other pillars are met, and the Malikis stipulate the guardian and witnessing (and witnessing at the time of consummation is sufficient). Therefore, your brother's relationship with that woman is not based on any legitimate religious evidence; rather, it is clear fornication (zina). He must be advised and made aware of the seriousness of what he is doing.