God as Trinity (Father, Son and Holy Spirit) cannot have a mother. Catholics have never understood the title “Mother of God” in that sense.
If we deny this title to Mary, however, are we also denying that Jesus Christ was divine? We may not make that link, but many people in the fifth century did. This title became official then as a way of settling that issue.
“Mother of God” is the popular translation of the title Theotokos (literally, "God-bearer"). A Concise Dictionary of Theology, by Gerald O’Collins, S.J., and Edward Farrugia, S.J., says that this title was used as early as the third century. The authors add, “When Nestorius of Constantinople called into question this popular title, the Council of Ephesus (431) condemned him and, in upholding the unity of Christ’s person, proclaimed the legitimacy of the title Theotokos.”
What is at stake here is not so much honoring Mary as acknowledging the uniqueness of Jesus—one person, who is fully God and fully human. Nestorius denied that Jesus was, in fact, one person and said that Mary should be called the Christotokos (Christ-bearer) but not Theotokos.