First, let’s make a distinction: Mary is not “responsible for Christ’s human nature” as if it were her direct creation, anymore than your mother is “responsible” for creating your human nature. Mary is Christ’s mother because God the Son incarnated in her womb by the power of the Holy Spirit. Analogously, your mother is your mother because God created you in her womb. Mothers are the mothers of persons, not natures. Christ is one divine Person who possesses two natures, one divine and one human. Because Christ is a divine Person (i.e., God), we can say that Mary is the Mother of God.
Moving on, Mary’s family is related to God by virtue of the Incarnation of God the Son, just as Mary is God’s Mother by virtue of the Incarnation. This means that it is indeed possible to say that St. Anne is the grandmother of God and that other blood relatives of Jesus through Mary are God’s blood relatives. That’s a consequence of the incredible mystery of the Incarnation: God loves us so much that he literally became a member of the human family.
