Using the new 12.0 Tor Browser for Linux the Tor process stays alive after the browser is closed.
Using version 11 of Tor Browser the Tor process ended automatically after closing the browser.
Older versions of Tor Browser which contained the same bug gave a helpful error message when you restarted Tor Browser which said that Tor was already running and must to be closed.
Version 12.0 of Tor Browser gives a more confusing error message when you restart the browser instead of directing you to close the Tor process:
Tor exited during startup. This might be due to an error in your torrc file, a bug in Tor or another program on your system, or faulty hardware. Until you fix the underlying problem and restart Tor, Tor Browser will not start.