Answer:
Mme is French, short for madame and equivalent to the English Mrs.
As it is French rather than English, it would mostly be used to give a "French touch", for example because the characters in a story are French.
See for example the Agatha Christie novels with detective Poirot - the author is English and the stories take place in England, but the character is consistently called "Monsieur Poirot" or "M. Poirot" to drive home that he is "foreign".
"Mme. Amalie was excited by her first trip to see the Eiffel Tower".