Not allowing people to call their custom docker tooling "docker-custom-tooling" on eg. Github seems like they're shooting themselves in the foot. Of course, it might be bad and damage their brand, but it could also enable a great ecosystem of surrounding tooling, some of which they could bless if they wanted to. Imho a poor PR move.
By no means am I a lawyer but I did a quick skim of their trademark agreement for fun:
Our Marks consist of the following registered, unregistered and/or pending trademarks, service marks and logos which are subject to change without notice: Docker, Docker “Moby Dock” whale logo and the stylist representations of a container, octopus, and other entities in our original works of authorship, DOCKER HUB, RUNHUB, LIBCONTAINER, LIBTRUST, LIBCHAN, LIBSWARM, LIBPACK, DOCKER SWARM, DOCKERIZE IT, CLAY
In this case, "Docker" is the trademark we're looking at, and specifically registered with a capital D it appears
Any permitted use must not falsely imply or suggest a sponsorship or endorsement by, or a partnership or affiliation with Docker.
I don't think the author is being malicious in any way of course but from a quick glance, I couldn't say for sure that it's not a Docker product (design guidelines aside)
The Docker mark is used only in a referential context or for naming Docker or to indicate compatibility, and not in a title of a program, domain name, website, product or service
This is essentially the key distinguisher between the parent comment and the OP's site
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Anyway, it's not so much that I'm pro-Docker or anything, it's more that they don't really have a choice legally. If companies don't protect the use of their copyrights/trademarks etc, they are diluted and risk the chance of becoming "generic". Having said that, it can definitely go overboard and lead to frivolous DMCAs, lawsuits etc etc
All this aside, while I'm not sure I have a use for the site (most things I dockerise use a compose file w/ multiple containers), it seems like a neat project so keep it up :)