Nicolas Sarkozy used this expression in 2012 after he was defeated in the 2012 election by current French president, François Hollande.
During a speech at the Mutuality in Paris on the 6th of May 2012, he said :
"Je m'apprête à redevenir un Français parmi les Français. Une autre époque s'ouvre, dans cette nouvelle époque, je resterai l'un des vôtres, je partage vos idées et vos convictions, et vous pourrez compter sur moi, mais ma place ne pourra plus être la même".Meaning, since he is no longer the French president, he is going to become "un Français parmi les Français". But we know this is not true. He will always remain a VIP and the press will always be interested in everything he does.
He cannot shop in Carrefour like the normal French people without anyone noticing that he's the Former French president. Neither can he walk on the streets without paparazzis falling over themselves to take pictures of him.
Bref, Nicolas Sarkozy will not and will never be "un Français parmi les Français".
Another similar expression is "un français moyen" which means an ordinary French person. To learn more about this expression, go here.