Keeping in mind european attitudes to carrying ID - in some EU countries there are formal and informal penalties if you don't have it - this is not especially far fetched.
It’s obligatory in France. Until recently the french ID was comically large which led to men having to carry massive wallets which gave rise to the ubiquitous “man bag”.
For people who don't read French, yes on paper there is no legal obligation to have any kind of ID on you.
However, police forces are allowed to ask anyone their identity without any reason, and if they can't justify of their identity, they can be detained by the police for up to 4 hours for verification, and Police may requires them to do a fingerprint scan: https://www.service-public.fr/particuliers/vosdroits/F1036
This is not correct, eventhough sometimes falsly claimed even by police officers.
You are obliged to posess an ID card once you turn 16 years old (§1 PersAuswG). But you don't have to carry it with you, eventhough it might complicate things e.g. should you be cought comitting a small crime.
You need to have it on you, or another valid ID, when crossing the border though.
In France, you must always carry a means of identification with you. However, a friend who is able to positively identify you is enough (legally - but obviously police will challenge you).
However, ID is never controlled, and many live with OQTF (Order to leave the French territory).