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Let's put it this way, you use brew install packagename to install command-line tools, and it's generally impossible to affect the graphical interface.
brew cask install packagename is possible.
mysql official documentation is installed through the dmg file:

The MySQL Installation Package includes a MySQL preference pane that enables you to start, stop, and control automated startup during boot of your MySQL installation.

this panel can only turn on, off, and control the boot self-boot mysql function. The mysql you install with brew install mysql can use Homebrew Services to achieve these functions.

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$ brew services run mysql
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$ brew services stop mysql
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$ brew services restart mysql
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$ sudo brew services start mysql