As with Airbnb everywhere, whether a French listing welcomes pets is the individual host's decision, stated in the listing's House Rules section. Many hosts in the French countryside and in cities with a strong café culture do welcome pets, but it's never guaranteed by default, and some specifically exclude them, particularly in shared apartment buildings or listings with delicate furnishings.
If the listing is silent on pets either way, message the host before booking rather than assuming. French hosts can and do decline or cancel a booking if an undisclosed pet turns up, and Airbnb's standard cancellation terms rarely favour the guest in that situation.
Ask specifically about any pet cleaning fee, whether the property has a secure outdoor space, and how close it is to a park or dog-friendly walking route — details that matter more in a self-catering stay than they would at a hotel with its own grounds.