Separation of sessions (cookies) for different mailboxes does not work. Windows 7, Firefox 61 (x64), x-notifier 4.2.3.
In the old version of x-notifier 3.5.23, the tab sessions were separated and the tab title was highlighted in underlined font. Previously, you could open different mailboxes of one mail service in different tabs. In version 4+, this does not work (triggered relogin).
Not currently...
Many features had to be dropped when Mozilla switched, and forced developers, to a different Application Programming Interface. However, Gmail supports multi-account access from one client... so X-Notifier 4.x can have several gmail accounts open in different tabs.
Who knows when they may change the API again.