Is it permissible for a non-Muslim woman to marry a Muslim man, knowing that she believes in the prophets and the Quran, but does not identify herself as Christian or Muslim, especially since the couple lives in a non-Islamic country and has two children? And is the husband obligated to separate from her despite the potential consequences for the children?

Civil marriage is valid if it fulfills the Sharia (definition) conditions and pillars; otherwise, it is null and void. There is no difference between a Muslim woman and a Kitabiyyah (woman of the People of the Book) regarding the necessity of fulfilling these conditions. However, if the woman refuses to adhere to the rulings of Islam after testifying that there is no god but Allah and that Muhammad is the Messenger of Allah, then she must be asked about her intention when she says she is not Christian. If she admits to having left the Christian religion, then she is an infidel (kafirah), not a Kitabiyyah, and it is not permissible to marry her. You must separate from her immediately, based on Allah's saying: "And do not hold on to the bonds of disbelieving women." But if she explains her statement in a way that indicates her continued adherence to Christianity, then it is permissible to marry her.