What is the validity of the belief in the existence of a hidden twin for a newborn, evidenced by the multiplicity of the umbilical cord, and is this considered polytheism (shirk)? What is the ruling on parents obeying the midwife by offering sacrifices or building a dwelling for this alleged twin?

This statement has no basis in Islamic law; it is one of the superstitions and falsehoods that rely solely on delusions, and it is not permissible to believe it or base rulings upon it. If the newborn recovers after the midwife performs what you mentioned, it might be a gradual enticement (istidraj) from Allah, as confirmed by His saying: "So leave Me and those who deny this discourse. We will gradually lead them [to ruin] from where they do not perceive," and His saying: "So when they forgot that by which they had been reminded, We opened to them the doors of every [good] thing until, when they rejoiced in what they had been given, We seized them unexpectedly, and suddenly they were in despair." It is not permissible to obey parents in an act of disobedience to Allah the Almighty.