https://mail.google.com/mail/u/2/#inbox
The "u/2" part indicates that this is the gmail view for the third (counting 0) user
For Google Apps for work, if you go to mail.google.com/a/example.com, it will let you log in separately to an example.com account and then redirect to a url like the one above.
FWIW: instead of manually entering the numeric index, you can enter the full email address instead. so you can enter /u/[email protected] or /u/[email protected]. This is handy because the numeric index depends on the order in which you logged into (added) the accounts and I don't want to have to remember that!
When a notification is received, the browser's Service Worker can send an "I got this" webhook event back to the sender. The sender can then decide if they wish to send a message to other browsers that user may have installed. Note that this does also require a user to be logged in to the site so they can be identified across both browsers.
Why have two browsers open at once? So that I can be logged into Google services simultaneously with two different accounts (personal and work). Off topic, but would be glad to know if there's another way to do that.