while docker does have a non-root installer, the default installer for docker is docker as root, containers as non-root, but since in order to manage docker as a whole it would need access to the socket, if docker has root the container by extension has root.

Even so, if docker was installed in a root-less environment then a compromised manager container would still compromise everything on that docker system, as a core requirement for these types of containers are access to the docker socket which still isn't great but is still better than full root access.

To answer the question: No it doesn't require it to function, but the default configuration is root, and even in rootless environment a compromise of the management container that is meant to control other containers will result in full compromise of the docker environment.