Can't install both docker CLI and Docker app

[osalbahr]

brew gist-logs <formula> link OR brew config AND brew doctor output

~$ brew gist-logs --cask docker
Usage: brew gist-logs [options] formula

Upload logs for a failed build of formula to a new Gist. Presents an error
message if no logs are found.

      --with-hostname              Include the hostname in the Gist.
  -n, --new-issue                  Automatically create a new issue in the
                                   appropriate GitHub repository after creating
                                   the Gist.
  -p, --private                    The Gist will be marked private and will not
                                   appear in listings but will be accessible
                                   with its link.
  -d, --debug                      Display any debugging information.
  -q, --quiet                      Make some output more quiet.
  -v, --verbose                    Make some output more verbose.
  -h, --help                       Show this message.
Error: invalid option: --cask
~$ brew gist-logs cask/docker
Error: No available formula with the name "cask/docker".
~$ brew gist-logs homebrew/cask/docker
Error: No available formula with the name "homebrew/cask/docker".
~$ brew config
HOMEBREW_VERSION: 4.0.15
ORIGIN: https://github.com/Homebrew/brew
HEAD: 763c41f006ab9b44261e313b0aaa24a047431b06
Last commit: 4 days ago
Core tap JSON: 28 Apr 07:20 UTC
HOMEBREW_PREFIX: /usr/local
HOMEBREW_CASK_OPTS: []
HOMEBREW_MAKE_JOBS: 4
Homebrew Ruby: 2.6.10 => /System/Library/Frameworks/Ruby.framework/Versions/2.6/usr/bin/ruby
CPU: quad-core 64-bit kabylake
Clang: 14.0.0 build 1400
Git: 2.37.1 => /Library/Developer/CommandLineTools/usr/bin/git
Curl: 7.87.0 => /usr/bin/curl
macOS: 12.6.5-x86_64
CLT: 14.2.0.0.1.1668646533
Xcode: N/A
~$ brew doctor
Your system is ready to brew.
~$ brew gist-logs docker
Error: No logs.
~$ 

Verification

• My "brew doctor output" says Your system is ready to brew. and am still able to reproduce my issue.
• I ran brew update and am still able to reproduce my issue.
• I have resolved all warnings from brew doctor and that did not fix my problem.
• I searched for recent similar issues at https://github.com/Homebrew/homebrew-core/issues?q=is%3Aissue and found no duplicates.

What were you trying to do (and why)?

~$ docker run alpine
docker: Cannot connect to the Docker daemon at unix:///var/run/docker.sock. Is the docker daemon running?.
See 'docker run --help'.

I know that I can run the daemon by opening the Docker app (open -a docker). So I wanted tobrew install --cask docker.

What happened (include all command output)?

Summary

~$ brew install --cask docker
==> Caveats
If your CLI tools were symlinked to $HOME/.docker/bin your path should be modified to include:

  $HOME/.docker/bin

==> Downloading https://desktop.docker.com/mac/main/amd64/106363/Docker.dmg
######################################################################### 100.0%
==> Installing Cask docker
==> Moving App 'Docker.app' to '/Applications/Docker.app'
==> Backing App 'Docker.app' up to '/usr/local/Caskroom/docker/4.19.0,106363/Doc
==> Removing App '/Applications/Docker.app'
==> Purging files for version 4.19.0,106363 of Cask docker
Error: It seems there is already a Binary at '/usr/local/share/zsh/site-functions/_docker'.

Full output

~$ /bin/bash -c "$(curl -fsSL https://raw.githubusercontent.com/Homebrew/install/HEAD/uninstall.sh)"
Warning: This script will remove:
/Users/osos/Library/Caches/Homebrew/
/Users/osos/Library/Logs/Homebrew/
/usr/local/Caskroom/
/usr/local/Cellar/
/usr/local/bin/brew -> /usr/local/bin/brew
Are you sure you want to uninstall Homebrew? This will remove your installed packages! [y/N] y
==> Removing Homebrew installation...
Warning: Failed to delete /usr/local/Caskroom
rm: /usr/local/Caskroom: Permission denied
Warning: Failed to delete /usr/local/Cellar
rm: /usr/local/Cellar: Permission denied
==> Removing empty directories...
Password:
==> /usr/bin/sudo /usr/bin/find /usr/local/bin /usr/local/etc /usr/local/include /usr/local/lib /usr/local/opt /usr/local/sbin /usr/local/share /usr/local/var /usr/local/Caskroom /usr/local/Cellar /usr/local/Homebrew /usr/local/Frameworks -name .DS_Store -delete
==> /usr/bin/sudo /usr/bin/find /usr/local/bin /usr/local/etc /usr/local/include /usr/local/lib /usr/local/opt /usr/local/sbin /usr/local/share /usr/local/var /usr/local/Caskroom /usr/local/Cellar /usr/local/Homebrew /usr/local/Frameworks -depth -type d -empty -exec rmdir {} ;
==> Homebrew uninstalled!
The following possible Homebrew files were not deleted:
/usr/local/Frameworks/
/usr/local/Homebrew/
/usr/local/bin/
/usr/local/etc/
/usr/local/go/
/usr/local/include/
/usr/local/lib/
/usr/local/mysql -> /usr/local/mysql
/usr/local/mysql-8.0.32-macos13-x86_64/
/usr/local/opt/
/usr/local/sbin/
/usr/local/share/
/usr/local/var/
You may wish to remove them yourself.

~$ /bin/bash -c "$(curl -fsSL https://raw.githubusercontent.com/Homebrew/install/HEAD/install.sh)"
==> Checking for `sudo` access (which may request your password)...
Password:
==> This script will install:
/usr/local/bin/brew
/usr/local/share/doc/homebrew
/usr/local/share/man/man1/brew.1
/usr/local/share/zsh/site-functions/_brew
/usr/local/etc/bash_completion.d/brew
/usr/local/Homebrew
==> The following new directories will be created:
/usr/local/Cellar
/usr/local/Caskroom

Press RETURN/ENTER to continue or any other key to abort:
==> /usr/bin/sudo /bin/mkdir -p /usr/local/Cellar /usr/local/Caskroom
==> /usr/bin/sudo /bin/chmod ug=rwx /usr/local/Cellar /usr/local/Caskroom
==> /usr/bin/sudo /usr/sbin/chown osos /usr/local/Cellar /usr/local/Caskroom
==> /usr/bin/sudo /usr/bin/chgrp admin /usr/local/Cellar /usr/local/Caskroom
==> /usr/bin/sudo /usr/sbin/chown -R osos:admin /usr/local/Homebrew
==> /usr/bin/sudo /bin/mkdir -p /Users/osos/Library/Caches/Homebrew
==> /usr/bin/sudo /bin/chmod g+rwx /Users/osos/Library/Caches/Homebrew
==> /usr/bin/sudo /usr/sbin/chown -R osos /Users/osos/Library/Caches/Homebrew
==> Downloading and installing Homebrew...
HEAD is now at 7386d4e40 Merge pull request #15316 from Homebrew/dependabot/bundler/Library/Homebrew/i18n-1.13.0
==> Installation successful!

==> Homebrew has enabled anonymous aggregate formulae and cask analytics.
Read the analytics documentation (and how to opt-out) here:
  https://docs.brew.sh/Analytics
No analytics data has been sent yet (nor will any be during this install run).

==> Homebrew is run entirely by unpaid volunteers. Please consider donating:
  https://github.com/Homebrew/brew#donations

==> Next steps:
- Run brew help to get started
- Further documentation:
    https://docs.brew.sh

~$ docker run alpine
-bash: docker: command not found
~$ file /usr/local/share/zsh/site-functions/_docker
/usr/local/share/zsh/site-functions/_docker: cannot open `/usr/local/share/zsh/site-functions/_docker' (No such file or directory)
~$ brew install docker
Warning: Treating docker as a formula. For the cask, use homebrew/cask/docker
==> Fetching docker
==> Downloading https://ghcr.io/v2/homebrew/core/docker/manifests/23.0.5
######################################################################### 100.0%
==> Downloading https://ghcr.io/v2/homebrew/core/docker/blobs/sha256:aa59d361b28
==> Downloading from https://pkg-containers.githubusercontent.com/ghcr1/blobs/sh
######################################################################### 100.0%
==> Pouring docker--23.0.5.monterey.bottle.tar.gz
==> Caveats
Bash completion has been installed to:
  /usr/local/etc/bash_completion.d
==> Summary
🍺  /usr/local/Cellar/docker/23.0.5: 12 files, 32.5MB
==> Running `brew cleanup docker`...
Disable this behaviour by setting HOMEBREW_NO_INSTALL_CLEANUP.
Hide these hints with HOMEBREW_NO_ENV_HINTS (see `man brew`).
~$ file /usr/local/share/zsh/site-functions/_docker
/usr/local/share/zsh/site-functions/_docker: ASCII text
~$ docker run alpine
docker: Cannot connect to the Docker daemon at unix:///var/run/docker.sock. Is the docker daemon running?.
See 'docker run --help'.
~$ open -a docker
Unable to find application named 'docker'
~$ brew doctor
Your system is ready to brew.
~$ brew install --cask docker
==> Caveats
If your CLI tools were symlinked to $HOME/.docker/bin your path should be modified to include:

  $HOME/.docker/bin

==> Downloading https://desktop.docker.com/mac/main/amd64/106363/Docker.dmg
######################################################################### 100.0%
==> Installing Cask docker
==> Moving App 'Docker.app' to '/Applications/Docker.app'
==> Backing App 'Docker.app' up to '/usr/local/Caskroom/docker/4.19.0,106363/Doc
==> Removing App '/Applications/Docker.app'
==> Purging files for version 4.19.0,106363 of Cask docker
Error: It seems there is already a Binary at '/usr/local/share/zsh/site-functions/_docker'.
~$ file '/usr/local/share/zsh/site-functions/_docker'
/usr/local/share/zsh/site-functions/_docker: ASCII text
~$ file $HOME/.docker/bin
/Users/osos/.docker/bin: cannot open `/Users/osos/.docker/bin' (No such file or directory)
~$ brew doctor
Your system is ready to brew.
~$ brew update && brew update
Already up-to-date.
Already up-to-date.
~$ brew doctor
Your system is ready to brew.
~$ time brew install --cask docker
==> Caveats
If your CLI tools were symlinked to $HOME/.docker/bin your path should be modified to include:

  $HOME/.docker/bin

==> Downloading https://desktop.docker.com/mac/main/amd64/106363/Docker.dmg
Already downloaded: /Users/osos/Library/Caches/Homebrew/downloads/a2fa53b312129a06f0f442d39dc52e4aa215dd8202dccd10ade7522d8fdd9584--Docker.dmg
==> Installing Cask docker
==> Moving App 'Docker.app' to '/Applications/Docker.app'
==> Backing App 'Docker.app' up to '/usr/local/Caskroom/docker/4.19.0,106363/Doc
==> Removing App '/Applications/Docker.app'
==> Purging files for version 4.19.0,106363 of Cask docker
Error: It seems there is already a Binary at '/usr/local/share/zsh/site-functions/_docker'.

real	0m59.864s
user	0m3.605s
sys	0m6.986s
~$ brew doctor
Your system is ready to brew.
~$ 

What did you expect to happen?

There shouldn't be an issue having both docker and the Docker app.

Step-by-step reproduction instructions (by running brew commands)

brew install docker
brew install --cask docker

[osalbahr]

I will move this to https://github.com/Homebrew/homebrew-cask/issues

Edit: moved to Homebrew/homebrew-cask#146078