Sometimes things get weird. One could imagine that the documentation on docker (https://docs.docker.com/engine/installation/linux/fedora/) could be up to date. And when it comes to installation itself, it actually is.

The problem is/was that I was forced to run the docker with sudo (reasons are explained on both pages linked here, I’m not going to repeat them), and while both sites gave a solution, the docs.docker.com -instructions did not actually work. So I googled a bit more:
https://developer.fedoraproject.org/tools/docker/docker-installation.html

According to developer.fedoraproject.org, you’ll have to run the following two commands in order to get docker executed without sudoing.
Basically you’ll add a docker -group and add yourself to it.

$ sudo groupadd docker && sudo gpasswd -a ${USER} docker && sudo systemctl restart docker
 $ newgrp docker