I have Python 3.8 and 3.9 installed via Homebrew:

~ brew list | grep python
python@3.8
python@3.9

I want to use Python 3.9 as my default one with python3 command. I tried the following:

~ brew switch python 3.9
Error: python does not have a version "3.9" in the Cellar.
python's installed versions: 3.8.6

I tried to uninstall Python and reinstall it, but it's used by other packages:

~ brew uninstall python
Error: Refusing to uninstall /usr/local/Cellar/python@3.8/3.8.6
because it is required by glib and php, which are currently installed.
You can override this and force removal with:
  brew uninstall --ignore-dependencies python

How can I use Python 3.9?

There is an Homebrew known issue related to side by side install of Python 3.8 / 3.9. To workaround, following commands should work for you:

brew unlink python@3.9
brew unlink python@3.8
brew link --force python@3.9

Re-opening your terminal or execute command rehash can be required to take account the change.