As long as the child is flying with the parent or guardian then they usually don't require any other identification.
But if the child is under 2 years of age and plans to fly free with a parent then you need to provide proof of that child's age with a birth certificate or some other proof that the child is under 2 years old to fly free.
Children 2 years and above will need to fly in their own seat and require their own plane ticket but you shouldn't need any ID for the child.