The official way to get informations about the owner of a given property is described here (in French) : http://vosdroits.service-public.fr/particuliers/F17759.xhtml
You are supposed to get informations from the cadastre that identify the property on a map with reference number.
Once you have this reference number, you need to ask to Service de publicité foncière who owns this reference number property. In the case of an apartment being part of a larger building, you will also need the lot number (which apartment in this given building). It doesn't seem you can do that online.
Depending on the size of the city where you are looking for this information, you might obtain this information by phone from the mayor's office (for small cities).
In general, it is not very common in France to ask a proof of property when dealing with an owner to rent a flat as an individual. There are some examples of scams where someone seems to be the owner of an apartment (has the keys, etc.) and ask you an advance fee in cash before disappearing. This is nevertheless not very common.
The common practice is to use basic clues about a potential scam : don't pay cash advances, try to get information on the property from neighbors, gardienne, etc. These are maybe not the easiest for a foreigner, unfortunately, so you might consider being helped by a local friend/colleague.
Details about relève cadastrale : there are two ways to query information. One way is to find the owner of a given building. The other way is to find all properties of a given owner. The cadastre website only allows to query by building address, and does not provide ownership information.