Do these profiles clear their cookies after request? I assume if the service finds a matching cookie, it will prefer it, or at least use as an extra identifier.
TEMP_DIR=$(mktemp -d /tmp/chromium.XXXXXXX) ; /usr/bin/chromium-browser --user-data-dir=$TEMP_DIR
At the end as other say they use hardware information + IP + other stuff. It is a lost battle.
Maybe these? https://browserleaks.com/webrtc But at least FF in private mode should randomize these IDs on restart.