Jewish father and Lutheran mother = ?

I’m writing a book character with a Jewish father and Lutheran mother. I myself am Lutheran, and am not entirely familiar with how religion is passed down in Judaism (I was told it generally follows a matriarchal line).

Would it make sense for this character to consider herself Lutheran and not Jewish? Like, she belongs to the Lutheran church and that is what she practices, though her family might celebrate both Christian and Jewish holidays?

How would this realistically look?

Please correct me on anything here!