and if I write: service s6 start (after stop), I can see I am still bound to console, which is for me as user not so clear why. For example nginx releases me back to console.
This is now a systemd question, and you should get help with systemd user groups, not here; but I see two things that seem incorrect in your service definition. I don't think "Type=forking" is correct, because s6-svscan doesn't fork; you probably want "Type=simple". And "StandardOutput=tty" looks suspicious. -- Laurent