It might be fair to say that the sense of SELF is an emergent property of this complicated interface but is in no way supernatural.
We don't know how it is we have a sense of self. It's a complete mystery. Consciousness is a mystery. To say that a brain is a computer is an unwarranted assumption.
Consciousness is not a mystery merely becuase you find it mysterious.
You speak of consciousness as though it were a minor little addition that the brain has and that computers do not have--that computers are just like brains--oh, except for this little matter of consciousness.