The problem was the missing gnome-keyring. Installation:

apt-get update
apt-get install -y gnome-keyring

And after a restart of the wsl the docker-credential-secretservice was running. Verification via docker-credential-secretservice list returns {}, which is a valid empty list. If you are on a interactive wsl you can create a keyring or if you not already have one you will be forced to do so by docker login by a little popup dialog.

In my case, during test automation I have to do it in a non-interactive and headless way. So the second part to get the problem solved is to create a dbus-run-session and pass the actions that need the login via script including the creation of a keyring.

dbus-run-session -- ./pull-docker.sh

and the pull-docker.sh contains:

#! /bin/bash

echo '$USER' | gnome-keyring-daemon --unlock
echo $GITHUB_TOKEN | docker login ghcr.io -u <your docker registry user> --password-stdin
docker compose pull