For example
chromium-browser --user-data-dir=/tmp/profile_A
chromium-browser --user-data-dir=/tmp/profile_A --incognito
chromium-browser --user-data-dir=/tmp/profile_B
chromium-browser --user-data-dir=/tmp/profile_B --incognito
For each command + its incognito it can detect them as separate profiles.
For ultimate privacy one needs to everytime launch browser with a new profile.