The Webstorage object exploits the "localStorage" javascript object. This storage is accessible from all pages/tabs in the same browser on the same domain, and have no expiration date.
This mean you should use some kind of prefix in from of all your variable to keep them from mixing between your games, if they are hosted on the same domain.
A simple way to do so is to create 2 functions in C2, a setStorage and a getStorage, that would take the key "highscore" as a parameter (in your example), and set/get in the storage using the key "application_prefix"&"highscore".
Application_prefix needs to be a constant here, for example the name of the game and an underscore.
Once you've got this, just change all the calls to the set/get you had by calls to these functions instead, and it should be all good.