It's not currently possible, but I believe it is possible in principle if the various Python formulae were fixed up to support this.

Dealing with the various executables (python3, pip3, etc) is fairly straightforward, but there are some conflicts that will require a bit more work:

❯ brew link --force --dry-run --overwrite python@3.10
Would remove:
/usr/local/bin/2to3 -> /usr/local/Cellar/python@3.9/3.9.9/bin/2to3
/usr/local/bin/idle3 -> /usr/local/Cellar/python@3.9/3.9.9/bin/idle3
/usr/local/bin/pip3 -> /usr/local/Cellar/python@3.9/3.9.9/bin/pip3
/usr/local/bin/pydoc3 -> /usr/local/Cellar/python@3.9/3.9.9/bin/pydoc3
/usr/local/bin/python3 -> /usr/local/Cellar/python@3.9/3.9.9/bin/python3
/usr/local/bin/python3-config -> /usr/local/Cellar/python@3.9/3.9.9/bin/python3-config
/usr/local/bin/wheel3 -> /usr/local/Cellar/python@3.9/3.9.9/bin/wheel3
/usr/local/share/man/man1/python3.1 -> /usr/local/Cellar/python@3.9/3.9.9/share/man/man1/python3.1
/usr/local/lib/pkgconfig/python3-embed.pc -> /usr/local/Cellar/python@3.9/3.9.9/lib/pkgconfig/python3-embed.pc
/usr/local/lib/pkgconfig/python3.pc -> /usr/local/Cellar/python@3.9/3.9.9/lib/pkgconfig/python3.pc
/usr/local/Frameworks/Python.framework/Headers -> /usr/local/Cellar/python@3.9/3.9.9/Frameworks/Python.framework/Headers
/usr/local/Frameworks/Python.framework/Python -> /usr/local/Cellar/python@3.9/3.9.9/Frameworks/Python.framework/Python
/usr/local/Frameworks/Python.framework/Resources -> /usr/local/Cellar/python@3.9/3.9.9/Frameworks/Python.framework/Resources
/usr/local/Frameworks/Python.framework/Versions/Current -> /usr/local/Cellar/python@3.9/3.9.9/Frameworks/Python.framework/Versions/Current

This might require special handling of the contents of *.framework directories in brew.

After installing python3 with brew install python3 I was getting the error:

Error: An unexpected error occurred during the `brew link` step
The formula built, but is not symlinked into /usr/local
Permission denied @ dir_s_mkdir - /usr/local/Frameworks
Error: Permission denied @ dir_s_mkdir - /usr/local/Frameworks

After typing brew link python3 the error was:

Linking /usr/local/Cellar/python/3.6.4_3... Error: Permission denied @ dir_s_mkdir - /usr/local/Frameworks

To solve the problem:

sudo mkdir -p /usr/local/Frameworks
sudo chown -R $(whoami) /usr/local/*
brew link python3

After this, I could open python3 by typing python3 👍

(From https://github.com/Homebrew/homebrew-core/issues/20985)

Did you try brew link --overwrite python?

brew install python or brew info python output mentions:

Unversioned symlinks python, python-config, pip etc. pointing to python3, python3-config, pip3 etc., respectively, have been installed into /opt/homebrew/opt/[email protected]/libexec/bin

So running, as root (for Bash (executable bash), use the ~/.bash_profile file):

echo 'export PATH=/opt/homebrew/opt/[email protected]/libexec/bin:$PATH' >> ~/.zprofile
source ~/.zprofile

gets you those symlinks created by Homebrew, python=python3, pip=pip3, etc.

python --version

Output:

Python 3.9.10

And:

pip --version

Output:

pip 21.3.1 from /opt/homebrew/lib/python3.9/site-packages/pip (python 3.9)

You can use the fat arrow here, so if you wanted foo to point to libexec/foo.sh then you could say:

bin.install_symlink "#{libexec}/foo.sh" => "foo"

Probably the safest and easy way is to use brew and then just modify your PATH:

First update brew:

brew update

Next install python:

brew install python

That will install and symlink python3 to python, for more details do:

brew info python

Look for the Caveats:

==> Caveats
Python has been installed as
  /usr/local/bin/python3

Unversioned symlinks `python`, `python-config`, `pip` etc. pointing to
`python3`, `python3-config`, `pip3` etc., respectively, have been installed into
  /usr/local/opt/python/libexec/bin

Then add to your path /usr/local/opt/python/libexec/bin:

export PATH=/usr/local/opt/python/libexec/bin:$PATH

The order of the PATH is important, by putting first the /usr/local/opt/python/libexec/bin will help to give preference to the brew install (python3) than the one is in your system located in /usr/bin/python