You might have to add python 3.8.1 to your PATH in your ~/.bash_profile and put it first so that it overrides previous installations. First find out where python 3.8.1 was installed, then add it to your path like this:

export PATH="/PATH_TO_PYTHON/:${PATH}"

Ok, thanks to @gromgit from Homebrew community discussion (https://discourse.brew.sh/t/how-to-default-python-3-8-on-my-mac-using-homebrew/7050)

Here is the solution:

$ brew info python@3.8
python@3.8: stable 3.8.1 (bottled) [keg-only]
...
==> Caveats
Python has been installed as
  /usr/local/opt/python@3.8/bin/python3
...
python@3.8 is keg-only, which means it was not symlinked into /usr/local,
because this is an alternate version of another formula.

If you need to have python@3.8 first in your PATH run:
  echo 'export PATH="/usr/local/opt/python@3.8/bin:$PATH"' >> ~/.bash_profile

For compilers to find python@3.8 you may need to set:
  export LDFLAGS="-L/usr/local/opt/python@3.8/lib"

For pkg-config to find python@3.8 you may need to set:
  export PKG_CONFIG_PATH="/usr/local/opt/python@3.8/lib/pkgconfig"

I will stick to python (v3.7.6) at this time and wait for seamless upgrade of v3.8.1 in the future releases.

Here is the solution:

If existing symlinks belong to python 3.7 you should unlink them:
brew unlink python

Basically all you need to do:
brew link --force python@3.8

OR force the link and overwrite all conflicting files:
brew link --force --overwrite python@3.8

OR if needed list all files that would be deleted:
brew link --overwrite --dry-run python@3.8

Thus you can switch to any python version available in the Homebrew repo.

Also check out this answer for pyenv usage

I have a company Mac with Python 2.7 preinstalled to run older software.

$ brew install pyenv (successful)
$ pyenv install 3.9.2 (successful)
$ python --version
Python 2.7
$ pyenv global 3.9.2
$ python --version
Python 2.7 

It still says Python 2.7, so, I did the following commands

pyenv init
eval "$(pyenv init -)"

$ python --version
Python 3.9.2

More details: Link