Homebrew's python formula is Python 3. To avoid breaking applications which expect the python command to run Python 2, brew install python does not add a python command, only python3. This is included in the caveats, visible in brew info python:
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
When using homebrew only, it's easy to get python to point to Python 3 by adding a line like this to one's shell configuration:
export PATH="/usr/local/opt/python/libexec/bin:$PATH"
So, one solution could be to link pyenv to that directory instead:
ln -s $(brew --prefix)/opt/python/libexec/bin ~/.pyenv/versions/3-brew
This would make python work. However, it means that python3.6 won't work, because that executable lives back in $(brew --cellar python)/3.6.5, so it's not a complete solution. I haven't yet figured out anything that preserves both behaviors without manually adding symlinks to Homebrew's installed Python.