It depends on how invested you want the player to be with the character. Depending on how the story or gameplay shakes out, you may finding your horror game protagonists being… disposable… so for some games it becomes less about the individual character themselves and more of a broad archetype or role that they play. Think of games where you play as characters defined simply by their namesakes - The Knight, The Doctor, The Jock etc.
If you want to do a deeper, more personal dive then you want to devote yourself to a single character to get the most emotional investment and attachment from the player.