Yes, it does belong to the government. Thanks for pointing that out, that's fascinating to investigate. As far as I can see:
Prior to 1789, Roman Catholicism was the state religion, and I think the church and state were so intertwined that the question of which one owned a property may not have been clearly defined.
During the French Revolution, Notre Dame was seized by by the new National Assembly and the church was made a branch of the government and subservient to it. Notre Dame was actually rededicated to the "Cult of Reason" and ransacked. It ended up being used as a warehouse for food storage.
In 1801, Napolean re-established Catholicism as the state religion and handed the wrecked building back to the church. It was later restored, I think at the King's expense.
In 1905 the French government once again declared France as a secular country. All church buildings were declared the property of the State, but were granted to the various churches for their use for free, though remaining the property of the government. I believe the government continued to pay for the maintenance of Notre Dame for many years, although recently maintenance was covered by charity.
So in summary, it's a government building and always has been, just with that ownership defined in different ways. And given the government seems to have largely paid for building, rebuilding and maintenance through much of history, that's actually understandable from a secular legal perspective, even though it disagrees with its dedication to God.
Today the government owns it and simply grants the Catholic church the right to use it. They could easily reassign it to a different group or a different religion entirely, probably completely within the existing 1905 law.