It's not the actual souls that gets you through the door, it's defeating the bosses. They just play it out that way since you get their souls by beating them anyway. The souls in your inventory are only used to forge weapons/equipment. They e do carry over, but you won't be able to open the Shrine Winter with those alone. You can consume those souls the moment you get them in your first playthrough, so if they were the key to the door then you wouldn't be able to progress through the game.