SStorage (remember) - v0
Looks really good. It's clear and easy to understand. Regarding this specific implementation, a couple of things that came to mind:


In setRemember(), unless you specifically only want to support the values 1 or 0, the code can be changed to:


this.clientStorage = remember ? this.localStorage : this.sessionStorage;

In setRemember(), when switching storage mechanisms it might make sense to copy the values out of the old mechanism into the new one so anything the user has previously set will be preserved. Similary, if the user is changing from localStorage to sessionSorage, would it make sense to also clear the old storage mechanism? If making either of these changes, the method could also first compare the new value against the old value and only continue if there is actually a change.
What is o used for in log() method? 
In the log() method, depending on on the utility you're using for logging, you might consider logging the entire object, at once, as many console / logging utilities will allow you to inspect objects. This can sometimes be more informative, although here it's just about whatever would be more convenient because all values will be strings, anyway. If you go this route, you could rename the current function to something like logEach(). 
In the execution of clear(), should you preseve the h_token key?


Taking a step back, here is an alternate API for the utilitiy. Just thinking out loud. 


ssn(key) - Retrieves a value. Synonymous with get()
ssn(key, value) - Sets a value. Synonymous with set()
ssn(obj) - Set all values from obj. Synonymous with setAll()
ssn.clear() - Unchanged.
ssn.clone() - Synonymous with getAll()
ssn.log() - Log the clientStorage object as a whole, as described above.
ssn.logEach() - Log each property and corresponding value, one at a time.
ssn.remember() - Retrieves the current remember setting.
ssn.remember(value) - Sets the remember value. Synonymous with setRemember()


One final thought you may want to implement some sort of 'namespace' type of functionality to be able to differentiate between settings saved for different users. I don't suggest this for security reasons, but more from a user-experience perspective. I think, with the way it's set up, multiple users on the same computer will have their settings merged, which may get confusing.