Yes I have seen it said that beaver is a “traditional” Russian delicacy, but I didn't know anyone who ate it. Maybe it was a delicacy if you find yourself lost and starving in the mountains. But these trends come and go I suppose.
Maybe it is influenced by some silly government plan to cull beavers. They are becoming a big problem along railway lines because they eat wires and even railway sleepers. So they are considered pests. Many years ago all kinds of agencies tried to encourage people to eat the wild goats that were also a big pest. They would try to invade people’s dachas and eat all the vegetables and fruit. But if you have never tasted goat meat you are not missing anything; You would need steel teeth; so people just used to shoot the goats on sight and leave them for the scavengers.
In terms of game, my father tended to stick to deer, or the small wild pigs that were in the forests. Sometimes rabbit or duck. Venison soylanka was a favourite with my family.