Children until 12 years are required to have a child passport unless traveling to a country where a passport is required for entry. For children between the ages of 12 and 16, it depends on where you go: If the child or the young person moves only in the Schengen area, the identity card will do. If you also plan trips beyond, the passport is certainly more universally usable.
If a minor child is not traveling with the parents, but alone or with another adult, you are advised to give your child a Declaration of consent in which the parent or guardian grants permission to travel. Declaration of content of the parents you find here: Einverständniserklärung für ein ohne Eltern reisendes Kind
Einverständniserklärung für ein mit nur einem Elternteil reisendes Kind