MySQL equivalent of PostgreSQL's psql \c command
In the context of MySQL, you cannot switch users. You can switch database and host.

According to the MySQL Documentation on the MySQL Client


  connect [db_name host_name]], \r [db_name host_name]]
  
  Reconnect to the server. The optional database name and host name arguments may be given to specify the default database or the host where the server is running. If omitted, the current values are used.


This looks similar to how PostgreSQL does it.