Since Mary is Jesus’ mother, it must be concluded that she is also the Mother of God: If Mary is the mother of Jesus, and if Jesus is God, then Mary is the Mother of God.
Although Mary is the Mother of God, she is not his mother in the sense that she is older than God or the source of her Son’s divinity, for she is neither. Rather, we say that she is the Mother of God in the sense that she carried in her womb a divine person—Jesus Christ, God "in the flesh" (2 John 7, cf. John 1:14)—and in the sense that she contributed the genetic matter to the human form God took in Jesus Christ.
The Nestorian claim that Mary did not give birth to the unified person of Jesus Christ attempts to separate Christ’s human nature from his divine nature, creating two separate and distinct persons—one divine and one human—united in a loose affiliation. It is therefore a Christological heresy, which even the Protestant Reformers recognized. Both Martin Luther and John Calvin insisted on Mary’s divine maternity. In fact, it even appears that Nestorius himself may not have believed the heresy named after him. Further, the "Nestorian" church has now signed a joint declaration on Christology with the Catholic Church and recognizes Mary’s divine maternity, just as other Christians do.
Since denying that Mary is God’s mother implies doubt about Jesus’ divinity, it is clear why Christians (until recent times) have been unanimous in proclaiming Mary as Mother of God.
Yes. These words have a meaning.
Like I have long said, "scratch a Maryphobe and find a deficient Christology." They go hand-in-hand.
If you have problems with calling Mary *mother of Jesus*, take it up with the Holy Spirit who breathed it out in His inspired Scripture.
There's the source of so many Catholic failures...