Wikipedia version of the story Dunya reads to the children at the beginning of the book.
https://en.wikipedia.org/wiki/Father_Frost_(fairy_tale)
“Dunya hesitated. The elder children looked at each other. In Russian, Frost was called Morozko, the demon of winter. But long ago, the people called him Karachun, the death-god. Under that name, he was king of black midwinter who came for bad children and froze them in the night. It was an ill-omened word, and unlucky to speak it while he still held the land in his grip. Marina was holding her son very tightly. Alyosha squirmed and tugged his mother’s braid.
“Very well,” said Dunya after a moment’s hesitation. “I shall tell the story of Morozko, of his kindness and his cruelty.” She put a slight emphasis on this name: the safe name that could not bring them ill luck. Marina smiled sardonically and untangled her son’s hands. None of the others made any protest, though the story of Frost was an old tale, and they had all heard it many times before. In Dunya’s rich, precise voice it could not fail to delight.”
Arden, Katherine. The Bear and the Nightingale: A Novel (pp. 4-5). Random House Publishing Group. Kindle Edition.
Katherine Arden So, Morozko is the name of the Russian Jack Frost, a winter demon who is sometimes benevolent and sometimes cruel. He features in multiple fairy tales. What I found interesting about this character though is he has his mythical roots in slavic paganism, as a dark god of winter and death called Chernobog. He evolved over the years from a pretty powerful deity to sort of a wicked fairy-tale creature, and finally (after some European influence) to Ded Moroz, the Russian Father Christmas. I found this journey (from wicked pagan god to giver of treats to children) absolutely fascinating, and I wondered what would go through such a character's mind as he was making that transition over the centuries. Anyway. Um, do Russians know of their pagan past? I certainly can't speak for the country as a whole. I know there are historians and casual enthusiasts who know a great deal about it, and there are definitely practicing pagans in Russia today. I would venture to say that the average Russian knows more fairy tales and fairy tale characters than they do Slavic deities and practices. If that helps :)