I am looking for data on French buildings on an address level, e.g. elevation, height, number of floors, square meters, type of building etc.
Buidings are available for Paris on the city opendata portal (with some wanted properties). Have a look on the following link :
From what I know, you will not be able to find a dataset on the whole France. OSM data could be interesting locally but not at such scale.
You may explore some other cities opendata plateforms.