Your child does not need a document to travel within the country before the age of 16. In Germany an identification card is compulsory from the age of 16 years. An identity card applies to all trips inland and to countries of the Schengen Agreement as a valid identity document.
It is possible to apply for a document called “Kinderausweis” (child’s identity card).
The normal Passport can be applied for regardless of age for children. Some countries require a passport for entry, for example into the USA. The passport must be requested at the local registration office. For persons under 18 years of age, however, the written consent of both parents or a power of attorney of the non-present parent is required.
You are advised to provide your child with a written allowance to travel alone if he or she does so. You should also state the starting point, the final destination and the reason for travelling.