A monitoring container is started with -v /var/run/docker.sock:/var/run/docker.sock. What does this give it?

It gives full control of the Docker daemon, which is effectively root on the host. Anyone who can talk to the Docker socket can start a privileged container. That container can mount the host's whole root filesystem.

The other options are wrong:
• "Read-only access to other containers' logs" is wrong. The socket accepts every Docker command, not only reads.
• "Nothing beyond what any container can do, because the socket is namespaced" is wrong. The socket is not namespaced. It talks straight to the host's daemon.
• "Access only to the containers in the same Compose project" is wrong. Docker has no per-project limits on the socket.