The couple cohabits in the Elysee Palace as a matter of right because the male partner Hollande is the constitutionally elected President of France. Trierweller, of course, lives in the reflected glory of her lover boy and enjoys every moment of it. The world may address her as the First Lady of France but there is no constitutional provision to confer that title on her.
Trierweller lives in the Elysee Palace because President Hollande, who never tied the nuptial knot, loves to have Valerie around to boost his morale as she has been doing all along.
We the votaries of the old school of thought here in India would not approve an unmarried couple sleeping together in the same bedroom of the Presidential Palace in Paris. It would not be okay in India.
Never mind, East is East; West is West: Twain shall never meet.
Wish the unwed couple a happy living.