If it were not so then the wars of succession would not have been fought. Nor would rebellions have been crushed, in some cases with the assistance of other nations. Now this does not mean that every nation has to recognize this government.
The Confederate States of America would more than likely exist today had the European Nations extended full recognition and not just belligerent rights.
Napoleon may have become the head of state by a coup, but it was the recognition of the fact by the other countries that kept him there.
They may not have liked the fact that he made himself the ruler, but he was still recognized as such; as these other nations made treaties, traded, etc., with his regime. If they did not recognize his rule then they could either ignore him, i. e. no treaties, no trade or go to war to remove him.
It appears (and here I stand to be corrected) that it was only in late 1813 that the coalition of united nations decided to remove him from the throne and either put forward their own candidate or recognize the right of the Bourbons to recover their throne. The fact that the coalition required his abdication would lend support to the fact that they did recognize his rule as legitimate then.
In 1815, he did usurp the throne and the united nations declared him an outlaw and went to war to remove him. They recognized the Bourbons as the legitimate rulers of France despite what the French army and a portion of its population desired.
Good hunting
Ron