In general if a tool requires root to run, of course a system administrator could make some changes to the system to allow a user to run it (adding the user to a privileged group, making the binary setuid, etc).

But that's not the point. The point is, if I'm on a system where I don't have the ability to make system-wide changes (run docker as root, add myself to a special group, etc), then I can't run docker on that system.