BrowserId didn't tie your identity to your browser. It tied your identity to your email address, and made it possible to have several identities to choose from.

The selling point was the browser integration. This would be especially awesome on mobile.

BrowserID wasn't intended to place your identity in the web browser, it was intended to place your identity manager/authentication agent in the browser. Your identity under BrowserID is whatever email address you wish to use, and BrowserID as a protocol is about A) how to verify that the user controls that email address, then B) how to cache that verification in the browser so that the user can easily reuse that email identity across multiple pages/sites.

The login flow for Persona/BrowserID is essentially "Which email would you like to use for this site?" and let you use the same email for just about all of them or keep multiple identities, as many identities as you have email addresses you care to use.