No, it creates a new session id for each browser/device.
Simply pulling the session id from the DB would be a small security flaw.
Invalidating all session IDs when logging out has its advantages. Imagine you logged into your account in a public place, left the place but forgot to log out.
Instead of waiting for the session to invalidate (i.e. keep me logged in for X minutes), you can simply login from another device (e.g. your mobile) and logout.
Brave is based on chromium, yes.
Almost all browser are based on chromium now. Basically it is firefox vs chrome/chromium now.