The cookie-based method sounds a logical solution, and should work, as far as I recall, on non-Firefox browsers as well. If it does not, please provide a sample.
Store/Retrieve are somewhat similar to cookies. They are used to access browser local storage, an HTML5 standard for storing client-side (= browser) data as key/value pairs, and being a browser-dependent mechanism it does not support user-switching of browser any more than cookies do.
There are also some significant differences between the two, one of which is that cookies are automatically added to every relevant server request (as determined by domain/path/expiration), and therefore easily accessible on the server side, whereas local storage is not. You can find more infomration by googleing for a comparison between local storage vs cookies.
For best results, use the Firefox browser..