After a completely clean re-try, it looks like there was a mix of settings from an older Dokku version.

Digital Ocean's 1-click Dokku droplet does not include the latest Dokku version. After an initial setup, I upgraded the version without any success to the deployment. As a last resource, I created a non-1-click droplet and then installed Dokku manually, everything works as expected.

There are already issues in the official repo, so nothing new to add.