brew install [email protected] can't connect to local MySQL server through socket

You probably have some directories left from another mysql installation, that were not removed.

I solved this by doing the following:

First uninstall mysql

brew uninstall [email protected]

Delete the folders/files that were not removed

rm -rf /usr/local/var/mysql
rm /usr/local/etc/my.cnf

Reinstall mysql and link it

brew install [email protected]
brew link --force [email protected]

Enable and start the service

brew services start [email protected]