If you’re required to have it and present it on demand, then almost everyone will carry it, and the tiny minority not carrying their card will be automatically suspicious.
It’s a very short step from there to simply requiring that it be on your persons.
Exactly this is the kind of fine details that a lawyer distinguished in the law.
So no. Absolutely no. You do not have to carry it with you. If it comes to the need, then maybe the police have to scort you to your home and you have to show the ID. But you are not requested to have it with you at all times.
Slippery slopes work exactly because people are incapable to predict the long term consequences for eroding safety standards and are at the same time arrogant enough to believe to be very much capable. Its cognitive biases at work in selecting some of the most reckless approaches for dangerous situations.
"I am not a lawyer" but:
"Deutsche im Sinne des Art. 116 Abs. 1 GG sind nach § 1 Personalausweisgesetz (PAuswG) verpflichtet, sobald sie 16 Jahre alt sind und der allgemeinen Meldepflicht unterliegen oder sich überwiegend in Deutschland aufhalten, einen gültigen Ausweis zu besitzen und ihn auf Verlangen einer zur Feststellung der Identität berechtigten Behörde vorzulegen sowie einen Abgleich mit dem Lichtbild des Ausweises zu ermöglichen."
You must either carry a national ID document or, if you are requested to identify yourself by the police, make it available to them in reasonable time on request (say, if you left it at home, show it to them at a police station the next day).