Short Answer
To make a clean install of Python 3.6.5 use:
brew unlink python # ONLY if you have installed (with brew) another version of python 3
brew install --ignore-dependencies https://raw.githubusercontent.com/Homebrew/homebrew-core/f2a764ef944b1080be64bd88dca9a1d80130c558/Formula/python.rb
If you prefer to recover a previously installed version, then:
brew info python # To see what you have previously installed
brew switch python 3.x.x_x # Ex. 3.6.5_1
Long Answer
There are two formulas for installing Python with Homebrew: python@2 and python.
The first is for Python 2 and the second for Python 3.
Note: You can find outdated answers on the web where it is mentioned python3 as the formula name for installing Python version 3. Now it’s just python!
By default, with these formulas you can install the latest version of the corresponding major version of Python. So, you cannot directly install a minor version like 3.6.
Solution
With brew, you can install a package using the address of the formula, for example in a git repository.
brew install https://the/address/to/the/formula/FORMULA_NAME.rb
Or specifically for Python 3
brew install https://raw.githubusercontent.com/Homebrew/homebrew-core/COMMIT_IDENTIFIER/Formula/python.rb
The address you must specify is the address to the last commit of the formula (python.rb) for the desired version.
You can find the commint identifier by looking at the history for homebrew-core/Formula/python.rb
https://github.com/Homebrew/homebrew-core/commits/master/Formula/python.rb
Python > 3.6.5
In the link above you will not find a formula for a version of Python above 3.6.5.
After the maintainers of that (official) repository released Python 3.7, they only submit updates to the recipe of Python 3.7.
As explained above, with homebrew you have only Python 2 (python@2) and Python 3 (python), there is no explicit formula for Python 3.6.
Although those minor updates are mostly irrelevant in most cases and for most users, I will search if someone has done an explicit formula for 3.6.