There was a lot of crap that went on with Church property and buildings and their governance after the French Revolution in 1789. Napolean entered into a deal with the Church in the early 1800’s which restored many of its rights and liberties, but not all. Not sure whether full control of Notre Dame ever came back. I know that Napolean was married there even though he was an atheist. Napolean had the right to approve the Archbishop of Paris and forced them to go along.
I guess I’ll have to look into the current situation.