It's property of the state, and the church doesn't receive financial support or tax breaks, but it's still in charge of maintaining the interior and furniture of the buildings they use. Some of the furniture is property of the state (what dates from before 1905, but now always everything) and some is property of the church.

However, this is also a historical monument and the state has to ensure that they are well-maintained (including parts that are privately owned) and not disfigured, and can grant financial support for restorations when needed.