To which country did 'Aesop's fables' belong?

Well, actually they "belong" to the world, they have been translated from the original Greek to Latin and now into most languages of the world including French, English, Russian, Japanese, Spanish, German and so many more.

Since they are attributed to Aesop, a Greek slave and storyteller from the 6th Century BC, you could suppose that Greece could claim the fables of Aesop "belong" to it, but the best answer is probably still that they belong to the world.