What are the rules to this "CheeseGame"?
Thanks to @npst for the tip on the Japanese manufacturer's name, which lead to contacting them directly and receiving a response on the rules of the game!

Here is what they wrote back:


  Thank you for your inquiry.
  The image you have attached is "Balance Game" of our product "Cheese Game".
  This is a wooden toy that also becomes an interior, and the contents are 7 pieces of cheese, 1 mouse and 1 base.
  
  The attached PDF data is a guidance sheet enclosed with the product.
  Stack the cheese and the rat on the base so as not to collapse.
  There is no particular rule.
  Please try various stacking methods and enjoy.


Here is the image that comes with the game to explain the rules (in Japanese). But the image at least suggests some pretty interesting stacking options.



Not sure what they meant by "also becomes an interior", maybe "interior decoration"? Because I have been playing with it as a desk toy and it works well for that.

So, not an actual 2 player game as I was hoping but fun none-the-less, and glad to have been able to solve the mystery. Thanks to all who attempted to track this down.