@civonelez,
Would be helpful to know what OS you’re using when you see this, as well as whether you have any extensions installed in the browser at this time?
Just so we’re all on the same page here, this is likely an issue caused by something going on in your browser profile or system. There is nothing in Brave that prevents you from staying logged into Google (other than the settings you mentioned previously) after closing and relaunching the browser.
Another test you can try is to create a new/fresh profile (Menu --> More tools --> Add profile) and test the behavior when using the new profile (that is, log into your Google account with all the appropriate settings configured, close the browser and relaunch to see if you’re logged out).