brew install python or brew info python output mentions:
Unversioned symlinks
python,python-config,pipetc. pointing to
python3,python3-config,pip3etc., respectively, have been installed into
/opt/homebrew/opt/python@3.9/libexec/bin
So running
% echo 'export PATH=/opt/homebrew/opt/python@3.9/libexec/bin:$PATH' >> ~/.zprofile
% source ~/.zprofile
# For bash use ~/.bash_profile.
gets you those symlinks created by Homebrew – python=python3, pip=pip3 etc
% python --version
Python 3.9.10
% pip --version
pip 21.3.1 from /opt/homebrew/lib/python3.9/site-packages/pip (python 3.9)