How about you provide a way to generate an additional URL which would be associated with your own anonymous account rather than with an individual game? When this other URL is opened in a different browser (perhaps on a different device) you could associate that new session to the same anonymous account.

This way, if I know I'm about to terminate my browser session, or if I just want to move to a different device, I could do so without making an account.

The only problem is how to present this in the interface without confusing users and generating unnecessary clutter. I would probably put it near account creation options as an alternative.