The EU pet passport is the standard document for moving a pet between EU countries, including into France, and it's only issued to pet owners resident in the EU, by an authorised vet in their home country. It records the microchip number and the full rabies vaccination history, and — unlike a one-trip certificate — stays valid for the pet's life as long as the vaccination record is kept current.
If you live outside the EU — the US, UK, Canada, Australia and most other countries — the pet passport isn't available to you at all, regardless of how long you plan to stay. You'll need a health certificate instead, confirming the same microchip and rabies vaccination requirements, arranged through an accredited vet shortly before travel. For US travellers, this certificate is completed by a USDA-accredited vet and endorsed by USDA APHIS; for UK travellers, it's the same Animal Health Certificate process used for travel to any EU country.
One detail worth double-checking if your trip includes more than one EU country: a single EU pet passport or health certificate generally covers travel across the whole Schengen area and the EU, so there's no need for a separate French document on top of it — unlike Switzerland, which sits outside the EU and sometimes needs its own paperwork.