FullStory doesn't let you change the identity of a user once they have been assigned a unique User ID using FS.identify. If you call FS.identify with a User ID value that is different from the currently-identified user, FullStory will recognize the new User ID and automatically split the session, creating a new session associated with the new User ID.
The User ID value in question refers to the first parameter of the FS.identify function (e.g. FS.identify(uid)). This is how it appears in the FullStory user interface:
What about User Variables?
Other User Variables set using FS.setUserVars are not static and can be changed as the user's information changes.
For example, you might update a user variable like lifetimeValue_real each time a user spends on your site, or update displayName and email if the user changes their contact information. However, as described above, you can't change their core User ID value.