What are applications of Coin Change problem?
This is going to sound very silly but, I once had to implement the coin change solution for the purpose of... distributing change! 

The application was a soda pop vending machine that could accept bills and coins and dispense coins.

The spec required the use of a least coin pay, plus an alternative algorithm should one of the coin tubes be empty and thus unable to dispense coins of that denomination.

The classic example was buying a 60 cent soda pop with a dollar (I know, pop was cheap back then ;-). This leaves 40 cents change, or in the US, 1 quarter, 1 dime, and 1 nickel for least coin pay. If the nickel tube were empty though, the machine would dispense 4 dimes.

I know to is not quite the lofty heights of academia, but it was an actual application of the problem's solution. On a further note, more modern coin changer mechanisms implement their own coin management so for those more capable peripherals, we would just let the coin mech figure out if change could be made. 

If not, the customer would get the dreaded NO CHANGE error message and the money would be returned.