Nope. You have to possess a valid ID or passport, but you are not required to carry it. Keeping it at home is perfectly fine.

Carrying it is practical and most Germans do carry their ID, but it's not a requirement.

What are the penalties for not doing so? I'm always amazed at the willingness of Europeans to follow rules like these, regardless of their impact on personal sovereignty. People in Finland were the most extreme example of this behavior that I've ever encountered. People would look like you murdered a child for jaywalking in Helsinki.