Mozilla implements it like cookies:
DOM Storage can be cleared via "Tools -> Clear Recent History -> Cookies" when Time range is "Everything" (via nsICookieManager::removeAll)
In DOM Storage it is not possible to specify an expiration period for any of your data. All expiration rules are left up to the user. In the case of Mozilla, most of those rules are inherited from the Cookie-related expiration rules. Because of this you can probably expect most of your DOM Storage data to last at least for a meaningful amount of time.
Chrome implements it like cache:
LocalStorage is Not Secure Storage
HTML5 local storage saves data unencrypted in string form in the regular browser cache.
On disk until deleted by user (delete cache) or by the app
As for a "replacement for the Cookie", not entirely
Cookies and local storage really serve difference purposes. Cookies are primarily for reading server-side, LocalStorage can only be read client-side. So the question is, in your app, who needs this data — the client or the server?