But how could it distinguish different profile directories, if they use the same settings. I would assume profile id, directories, or others should not be exposed through the browser. I am not used to chromium-browser (is this chrome? forgive my incompetence), but I wonder what kind of profile-specific static identifiers despite cookies could leak out the browser?
Maybe these? https://browserleaks.com/webrtc But at least FF in private mode should randomize these IDs on restart.