Is New Year's Day a Catholic holiday?

Yes, but only because it is the solemnity of Mary, Mother of God, on the Church’s liturgical calendar. On many civil calendars, those which adopted the Gregorian Calendar, it is also New Year’s Day.

Why is Mary called the Mother of God?

The Catechism of the Catholic Church (paragraph 495) teaches,

Called in the Gospels "the mother of Jesus," Mary is acclaimed by Elizabeth, at the prompting of the Spirit and even before the birth of her son, as "the mother of my Lord." In fact, the One whom she conceived as man by the Holy Spirit, who truly became her Son according to the flesh, was none other than the Father's eternal Son, the second person of the Holy Trinity. Hence the Church confesses that Mary is truly "Mother of God" (Theotokos).