Bam, got it. The error you are seeing is actually to reference what is arguably either a webkit bug that's never been patched or a poor design choice by their devs. You see, webkit's incognito feature follows a philosophy that there should be no footprint on the disk correlating to that browsing session. As a response, their developers chose to disable localStorage and apparently sessionStorage as well. And for some reason, webkit's incognito mode behaves in a way that localStorage and sessionStorage will still appear in the global namespace, but it will become read-only, and thus acceps no quota. As a result, the script is confirming that the phone supports localStorage, but when it calls setItem(), it throws an exception. Then Macil wrote the script to potentially catch it, and calls alert() and prints out the error thrown.
Incidentally, there's an even more bizarre bug whereas when removeItem() and clear() are called, the browser doesn't even throw an exception. It just silently ignores the call. Therefore there is also another bug where if you uncheck a box in the setting, nothing happens.
That being said, this bug theoretically can be replicated on incognito for most webkit browsers, including Safari, both the iOS version and the desktop version. However, all versions of Google Chrome past 2010, whether webkit based have had it patched.
tl;dr - you need to find another mobile browser.