As both Abrahamic religions, Islam and Judaism have the same God and share a common law then how do rabbis see the marriage of a Jewish woman marrying a Muslim man and vice versa? What about their children? Would they be Jewish as they have a Jewish mother? Please comment and explain.
For many purposes of Jewish law, Muslims are treated exactly on a par with members of other faiths.
Judaism doesn't recognize intermarriage - in either direction - as valid (Code of Jewish Law, Even Haezer 44:8); indeed, a Jewish man would have to be prepared to give up his life rather than have sexual relations with a non-Jewish woman (Shach, Yoreh De'ah 157:12). The same rules, then, apply regardless of the non-Jewish partner's nationality or religion.
As for any children from such a union, the usual rule applies: if the mother is Jewish then they are too, and if not, they are not.
This requires making a distinction between Jewish peoplehood & religion and the Jewish faith:
The Jewish faith is a universal one: strict monotheism combined with a basic set of moral & societal rules (the sheva mitzvos b’nei Noach, the seven Noachide Laws). Any human being that follows these guidelines is on the correct path. Islam has no special status here, except to the extent that it (and Unitarian Christianity) also teaches strict monotheism.
The Jewish people have a special role in this scheme; we are bound by a special Covenant with God and have many additional laws to follow. Anyone can choose to join us and take on these obligations, but once accepted these obligations may not be put down. (This is one reason Judaism discourages converts: it’s much easier to be a righteous non-Jew than an observant Jew.)
That said, marriage between a Jew of either sex and a non-Jew is not allowed. Husband & wife must be bound to the same covenant with God: either the general one for all humanity or the particular one for Jews.
In many circumstances, a non-Jew can convert to Judaism, joining the Jewish people, and marry a born Jew. Such conversions are often suspect, though, with the convert’s true commitment (and hence the validity of the conversion) sometimes in question. Many Rabbis will therefore be reluctant to convert someone who is in a pre-existing relationship with a Jew.
In case of a violation of these rules, when a Jew & a non-Jew have children together, Jewish Law considers the children to inherit their mother’s status: if she is a Jew, so are her children, and vice versa.