Just tried this in Opera. Having never visited the site before, I opened it in a "New Private Tab". Set Opera to reject all [normal] cookies from that domain. Saw an ID number on the page; recorded it. Opened another private tab. Saw a different ID number. Refreshed page; got yet another (different) ID number. Revisited page within same [private] tab (pressed Enter in address bar): got yet another (different) ID number. Did the same in Chrome (regular tabs): saw same behaviour.

Using two different private tabs in Opera, I get two different IDs to start with, but when using the "click to rediscover" buttons, both allegedly private tabs [eventually] end up with the same ID.