Docker is built with Moby, but you don't need Moby to install the built version. It is still open source, and still free. In the terms and conditions you linked to, definitions 1.5 and 1.6:
1.5 “Licensed Software” means the Docker software identified on an Order Form (other than Open Source Software) and licensed to Customer pursuant to the terms of this Agreement, e.g., the “Docker Hub Enterprise” software or other licensed software from Docker that is identified on the Order Form (excluding any Open Source Software included therein).
They're being careful to exclude the open source version of Docker as "licensed", and
1.6 “Open Source Software” means Docker or third party software identified at https://www.docker.com/components-licenses, that is distributed or otherwise made available as “free software”, “open source software” or under a similar licensing or distribution model..
They're making sure that Docker is acknowledged as open source software.
I think the proprietary look is that they're trying to use one license to cover both the open source and the subscription products. It would be clearer if they split the licenses in two.