Recent winners in presidential elections in France have been free to form their own government, because, thanks to a forced synchronization of the legislative and presidential elections by former French President Jacques Chirac, the legislative elections are two months after the presidential ones. So the governments formed in this period don't last long in any case.
However, nobody talks about the legislative elections in the movie, which spans several months, which leads to believe that the two elections are unsynced in the in-movie universe. In this case, the winner of the presidential election doesn't have much choice but to nominate the leader of the majority party in the National Assembly as Prime Minister. A different choice would most likely lead to a successful vote of no-confidence in the government by the National Assembly. See more »