I believe no. I'm running Firefox with arkenfox user.js and when I take this test https://www.bromite.org/detect it shows a new and different fingerprint if everytime i close and reopen the browser. Feel free to try it for yourself.
And while Brave may be private from outsiders, it is far from private from Brave Software themselves and I wouldn't trust them if I was honest with you. If you want an alternative chromium based browser, check out Vivaldi. They don't have aaaas many privacy features built in as Brave does but you can still get very private and obviously tack on Ublock origin and a customized DNS block list like you normally would with any other browser. And they are significantly more trustworthy than Brave
