Isn't mandated child support basically just welfare for needy children?

What is the moral difference between compelling parents to support their children and compelling all people to support the needy in society? Many critics of the welfare state believe that parents should be compelled to support their children with basic levels of physical sustenance and education, such that failing to provide these constitutes violating children's rights. But how is that different from compelling people to support other needy or vulnerable people? Is the blood relationship what creates the obligation to support the child -- and if so, how?