This is not a unique plot point from the movie, as it also appears in the source material (Charles Perrault's retelling of the classic folk take). 

In all versions of the story that I am aware of, Cinderella leaves some item of jewellery or clothing behind, and she can later be identified by the fact that she is the only person that the item fits. Some authors have the slipper being gold, or leather, and there are even versions where it is a ring that was left behind.

So, the out-of-universe explanation is that it's true to the original folk tale to have Cinderella being identified by means of her lost slipper.

I'm not aware of any in-universe explanation for the slipper not vanishing after midnight. However, I like to think that the fairy godmother's magic can last for a long as she wants it to. Perhaps the midnight curfew was an artificial limitation, designed to get Cinderella to leave the ball, making her more mysterious and desirable to the prince.