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Map of Île-de-France
What could the Island of France possibly be? It doesn’t make any sense. The entire country is attached to the mainland of Europe.
Well, the Île-de-France is actually one of the 21 administrative areas of France and contains a measly 2.2% of the French landmass and 17% of nation’s population. Paris lies squarely in the middle of the Île-de-France. This region has dominated French culture, history, and language. The design for the Gothic cathedral came out of this area. The French language came out of this area. Brie cheese came out of this area. And, I am sure, many other French things.
Detail map of the Ile-de-France
So, why again is a landlocked province in France called an island? As near as I can tell “this area was known as the ‘Ile de France’, or ‘isle of the Franks’ because it is virtually surrounded by tributaries of the Seine and because it had always been the center of Frankish rule” (source). In the middle ages, there was no sense of nationality. The only people who called themselves French were people from the Île-de-France.
In short, “island” because of the many rivers that surround the area made it feel hemmed in by water; “France,” because it was the origin of Frankish culture and rule.
The Reformed Francophobe 
Ahahah c’est vrai que ca peut paraitre dingue. Actually! I didn’t even exactly know why we are saying Ile de France. Now, I know 😀