It doesn't have to: they claim descent from Abraham, therefore what Abraham's God has already said doesn't need to be said again. But like you, they choose what they accept and what they don't accept, and interpret it to please themselves.
Yes it does have to. Claims of descent from Abraham does not mean adhering to the same ideology that was developing from Abraham. The descendants of Esau (Abraham’s firstborn grandson from Isaac) certainly did not; nor did the descendants of Ishmael, who most Arabs claim descent from.