What is the MySQL monitor? Upon a successful connection to the MySQL server, the client is greeted with "Welcome to the MySQL monitor". A web search returned many results on monitoring for MySQL databases. Is "monitor" here another name for "server"?
It is sometimes called the "commandline mysql tool".  It is a client that comes with MySQL and MariaDB.  It is a simple but very useful way to type SQL and get it executed.  Think of it as the "shell" version of phpmyadmin or Workbench.

It does not do any monitoring, though it could be used as a way to execute SQL in a monitoring script.