Once one is within the Schengen union. Is it necessary to have one's passport crossing from one Schengen country into another? For example, a Paris tour bus provides a Brussels guided tour for the day.
In reality, no, but countries laws change on what documents you require to be lawfully in the country, for example, in Italy you must have your passport with you.
However, crossing from France to Belgium, you will not encounter a regular border check. I would always keep my passport with me while travelling cross-border. I’ve never crossed the French and Belgian border, I have crossed the French-Italian border hundreds of times, and a good amount of those times there have been suprise identity checks.
Best advice is better to be safe than sorry and carry a passport cross-border every time.