The US Department of Homeland Security provides online access to arrival and departure date records of nonimmigrant aliens entering and departing the United States.
Do some French agency provide the same for people entering and departing France?
This information is not recorded. To boot, you are not crossing the French border, you are crossing the Schengen border. From an EU memo:
There are already measures and tools available at EU border crossing points, such as the Schengen Information System (SIS) and Visa Information System (VIS). However, none is intended for the purpose of recording border crossings