Docker in docker service broken

I read all the issues, but nothing helped me.

Summary

CI job uses docker:dind image as a service for a specific build. However, in execution I get the error

Steps to reproduce

.gitlab-ci.yml

image: tmaier/docker-compose:latest

services:
  - docker:18.09.7-dind

variables:
  DOCKER_DRIVER: overlay2

stages:
  - test
  - build
  - push
  - clean-up

before_script:
  - docker info
  - apk update
  - apk add make
  - make down

test:
  stage: test
  tags: [test]
  image: tiangolo/docker-with-compose:latest
  script:
    - make prod-up
    - make composer-init-dev
    - make test

build:
  stage: build
  script:
    - docker build -t $CI_REGISTRY/$CI_PROJECT_NAMESPACE/$CI_PROJECT_NAME:${CI_COMMIT_REF_NAME}_web-stack -f ./app/web-stack.prod.docker ./app

push:
  stage: push
  before_script:
    - echo "$CI_REGISTRY_PASSWORD" | docker login -u "$CI_REGISTRY_USER" "$CI_REGISTRY" --password-stdin
  script:
    - docker push $CI_REGISTRY/$CI_PROJECT_NAMESPACE/$CI_PROJECT_NAME:${CI_COMMIT_REF_NAME}_web-stack

clean-up:
  stage: clean-up
  script:
    - make down
  when: always

Actual behavior

docker:dind service fails to start

Running with gitlab-runner 12.6.0 (ac8e767a)
  on My Docker Runner FRft8kuk
Using Docker executor with image tiangolo/docker-with-compose:latest ...
Starting service docker:dind ...
Pulling docker image docker:dind ...
Using docker image sha256:66dc2d45749a48592f4348fb3d567bdd65c9dbd5402a413b6d169619e32f6bd2 for docker:dind ...
Waiting for services to be up and running...
Pulling docker image tiangolo/docker-with-compose:latest ...
Using docker image sha256:376b83c9b8a8b8972aafc9f7d4d7f29a8c6464bd201e7e826eb4c39fb52f2818 for tiangolo/docker-with-compose:latest ...
Running on runner-FRft8kuk-project-11-concurrent-0 via srv-w-gitrun01...
$ git config --global http.sslVerify false
Fetching changes with git depth set to 50...
Reinitialized existing Git repository in /builds/###/docker.###/.git/
From https://gitlab.###/yellowteam/docker.###
 * [new ref]         refs/pipelines/636 -> refs/pipelines/636
   6a063e5..1341d9d  master             -> origin/master
Checking out 1341d9d4 as master...

Skipping Git submodules setup
$ docker info
Client:
 Debug Mode: false

Server:
ERROR: Cannot connect to the Docker daemon at tcp://docker:2375. Is the docker daemon running?
errors pretty printing info
ERROR: Job failed: exit code 1

Expected behavior

Test, build and push

Environment description

sudo docker info

Containers: 2
 Running: 0
 Paused: 0
 Stopped: 2
Images: 55
Server Version: 18.09.1
Storage Driver: overlay2
 Backing Filesystem: xfs
 Supports d_type: true
 Native Overlay Diff: true
Logging Driver: json-file
Cgroup Driver: cgroupfs
Plugins:
 Volume: local
 Network: bridge host macvlan null overlay
 Log: awslogs fluentd gcplogs gelf journald json-file local logentries splunk syslog
Swarm: inactive
Runtimes: runc
Default Runtime: runc
Init Binary: docker-init
containerd version: 9754871865f7fe2f4e74d43e2fc7ccd237edcbce
runc version: 1.0.0~rc6+dfsg1-3
init version: v0.18.0 (expected: fec3683b971d9c3ef73f284f176672c44b448662)
Security Options:
 apparmor
 seccomp
  Profile: default
Kernel Version: 4.19.0-6-amd64
Operating System: Debian GNU/Linux 10 (buster)
OSType: linux
Architecture: x86_64
CPUs: 2
Total Memory: 1.947GiB
Name: srv-w-gitrun01
ID: 3RLU:EW67:4GFN:3MMD:47EA:A2YI:622C:6ZHB:3IFW:LVOA:H747:3ZHY
Docker Root Dir: /var/lib/docker
Debug Mode (client): false
Debug Mode (server): false
Registry: https://index.docker.io/v1/
Labels:
Experimental: false
Insecure Registries:
 127.0.0.0/8
Live Restore Enabled: false

config.toml contents

concurrent = 1
check_interval = 0

[session_server]
  session_timeout = 1800
[[runners]]
  name = "My Docker Runner"
  url = "https://###/"
  token = "FRft8kukkz3qwXw6kkFJ"
  executor = "docker"
  pre_clone_script = "git config --global http.sslVerify false"
  [runners.custom_build_dir]
  [runners.docker]
    tls_verify = false
    image = "tiangolo/docker-with-compose"
    privileged = true
    disable_entrypoint_overwrite = false
    oom_kill_disable = false
    disable_cache = false
    volumes = ["/cache"]
    shm_size = 0
  [runners.cache]
    Insecure = false
    [runners.cache.s3]
    [runners.cache.gcs]

Used GitLab Runner version

Version:      12.6.0
Git revision: ac8e767a
Git branch:   12-6-stable
GO version:   go1.13.4
Built:        2019-12-22T11:55:34+0000
OS/Arch:      linux/amd64