Well, there’s this in that document:
docker service starts automatically on Debian based distributions. On
RPM based distributions, such as CentOS, Fedora, RHEL or SLES, you need to start it manually using the appropriate
service command. As the message indicates, non-root users cannot run Docker commands by default.”
Perhaps it would be better if that said, “should start automatically”, as it doesn’t start for you. As for why it does not, we need to know first what mechanism on your Ubuntu 14 manages such auto-start apps, which are generically known as" services".
It would be useful for you to see this blog post which explains the 3 ways that might work (system d, upstart, and system v), as has varied over different distros and even different Ubuntu versions: https://linoxide.com/enable-disable-services-ubuntu-systemd-upstart/
service docker status
systemctl status docker
initctl status docker
Once we know which it is your system is using, we can then focus on why it’s not starting on reboot.