The Eastern Orthodox rites do not celebrate Christmas in the same way the Roman Catholic influenced world does. I cannot specify what happened in Russia, but the giving of gifts is instead on the Epiphany, traditionally 12 days after Christmas day (thus the "12 days" are actually a Catholic/Orthodox extension of Christmas that happen after the traditional birthday of Jesus.) The reason behind gift giving on Epiphany is because this day celebrates the Wise Men giving a young Jesus gifts. As the wise men (from an unspecific eastern country) gave Jesus gifts, Eastern rite Christians exchange presents with each other.

Gift giving is mostly done on New Years not Christmas. Christmas is celebrated on January 7.