In Judaism, and most sensibly, jewish lineage is traced through the mother, not the father – we all know who our mother is, not all of us know our father. ; )
In a patriarchal society, this produces a paradox, the man owns the woman, therefore the children are his but are really only his wife’s. It is thus that, assuming no adultery, (as this would produce a flawed line) the male line is named but, it should be the female line.
I assume that Christians follow the paternal line because that is what they do: they would be wrong. Christ came to save God’s Tribe, The Jews, not the rest of us.