I am French, and my mother used to cook delicious soupe de poissons...but still one of the best complementary cheese for any soup is aged parmesan...
but french people are more likely to use their own cheeses (believing that their cheeses are the best in the world, and the only ones worth using), so, no, not likely to be Emmental since that´s more of a Swiss cheese (though I have seen it used as well as Dutch Edam)
more typically gruyere is used since some of it is produced in France and is much more available in local supermarkets, but sometimes they use other, often soft-paste french cheeses for the soup including all the classics for a cheese fondue savoyarde