Solved. Solution:

In case someone finds himself in the same situation, I just followed the installation for NVIDIA Docker. It seems Docker 19.03 does not really work with GPUs by itself as I misunderstood... At least not for me. For convenience, these are the installation commands for Ubuntu/Debian:

# Add the package repositories
distribution=$(. /etc/os-release;echo $ID$VERSION_ID)
curl -s -L https://nvidia.github.io/nvidia-docker/gpgkey | sudo apt-key add -
curl -s -L https://nvidia.github.io/nvidia-docker/$distribution/nvidia-docker.list | sudo tee /etc/apt/sources.list.d/nvidia-docker.list

sudo apt-get update
sudo apt-get install -y nvidia-container-toolkit
sudo systemctl restart docker