If multiple databases are active for a user, the user can be switch between these databases via the Omnibar, which is the top bar containing the product, database name and the user's name.  
Omnibar

 

Please review the following browser behaviors and keep these in mind when accessing multiple databases:
  • To access multiple FE databases at the same time from your browser, you must be logged in as the same user (who has access/an auth ID to those databases). 
  • You cannot be logged in as different users on different tabs of the same browser.
  • When you sign in to the first tab of a browser, every tab afterward points to the same tenant/user combination. You must change databases from each tab.
  • When you sign out of one tab in a browser, you automatically sign out of all of them. Therefore, you cannot log out of one database on one tab, without logging out of all the tabs.  
  • You can be logged in as different users in different browsers (Chrome = user A, Firefox = User B).