Iām guessing that aws s3 sync command just syncs the local dir jekyll/_site/docs to the remote s3://circle-production-static-site/docs/, probably like (or using) Rsync.
I am sure you can deploy a Docker image. Assuming you have built one in your config, you can docker login to your registry, tag and push. I guess from there you can run a remote command on your deployment target to pull and restart the container, but it depends what you are using remotely (Docker, Docker Compose, Swarm, Kube, etc).