If you edit a product category by double clicking in the "site" tree, the edit will show in the site tree but not in the corresponding item dialog. This in turn leads to the menu on the sample site working, but if you click on such a menu item, it will not show any products associated with the product category (because in fact there are none given that the names don't match?).

Either disable editing directly in the site tree or ensure that editing will not negatively effect the product listing.