Indeed I stand corrected. Interestingly, (most) ancient cathedral are owned by the state, while (most) other churches are owned by the cities.

I made a bit of research, and while the sources are somewhat contradictory - the article linked as a source by Wikipedia contradicts what is stated on Wikipedia! - my understanding is that you are correct: - nationalized in 1789 - the Concordat (Napoleon, 1801) confirms the state ownership, and usage is granted to the Church - a few laws in the 19th century states the state pays for the maintenance of the building - law in 1905 confirms (again) state ownership, usage assigned to the Church