What I said applies to French citizens, not foreigners. For foreigners the situation is less clear, in some cases you need to be able to show your passport and visa http://vosdroits.service-public.fr/particuliers/F31208.xhtml -- if you are within the scope of a visa exemption I would imagine that you would need to justify it by producing a passport of the correct nationality and supporting documents if any.

(Personally, I always keep my passport with me when travelling abroad.)

In France I always have my ID with me but I know friends who don't (e.g., did not bother replacing a lost ID). Having lived 24 years in France I have never been asked to produce ID by the police or gendarmes. (That said, I am white, not often involved in demonstrations and such, etc.; other people's experiences may vary. :-/)