It has a variety of dates.
Do you know anyone who says: "I have to go Holiday shopping"? I don't.
Actually, Christmas has only one date.
December 6 (not 5th) is St. Nicholas' Feast Day, and some people exchange gifts that day. Children put their shoes out in hopes that St. Nicholas will fill them with candies and small trinkets.
Epiphany (January 6) is celebrated in Latin countries with gift exchanges as "Three Kings Day", for it is the day the Magi arrived to visit the Christ child and bring Him the traditional gifts of gold, frankincense and myrrh. It is the last of the "Twelve Days of Christmas".
So yes, there are other religious feast days during the season which are celebrated with gift giving but Christmas is December 25 and no other date.