The feast is the same mystery
The Church is celebrating the birth of Christ, the Word made flesh. The date difference is a calendar question, not a different doctrine about Christ.
For there is born to you this day in the city of David a Savior, who is Christ the Lord.
The Coptic date should therefore be read from inside the Church’s calendar. Kiahk 29 is the feast day; January 7 is how that date usually appears on the modern civil calendar. In some leap-year relationships the civil date can shift, but the liturgical anchor remains Kiahk 29.
Kiahk and the Nativity Fast
The month of Kiahk prepares the Church for the Nativity with hymns, praises, and readings centered on the Incarnation and St. Mary. The Nativity Fast gives this preparation a bodily form, so the feast is received with prayer and thanksgiving rather than treated as a calendar curiosity.
Kiahk also gives the Nativity a distinctive Coptic sound. The month is full of praise, especially hymns and readings that confess the Word of God taking flesh from the Virgin Mary. The date is therefore tied to a whole season of worship, not only to a disagreement about calendars.
How to understand the date
The helpful distinction is between the mystery and the calendar expression. The mystery is the Incarnation of the Son of God. The calendar expression is the Coptic Church’s received way of celebrating that mystery on Kiahk 29. Parishes usually have a late-night or evening Nativity Liturgy, with local timing announced by the parish.