How to expire user registration?

You would need to do a Cron.
Get all user who where registered before 6 months ago and change their role to something else. in this example i have user ‘expired’ but you would need the create this role before you could use it.

add_action('wp', function() {
   if ( !wp_next_scheduled( 'check_for_expired_users' ) ) {
      wp_schedule_event( time(), 'daily', 'check_for_expired_users');
   }
});

function check_for_expired_users() {
  $users = get_users([
    'role__not_in' => array('expired'),
     'date_query' => array(
       array(
         'before' => '6 months ago',
       )
    )
  ]);

  foreach ($users as $u) {
    $u->set_role( 'expired' );
  }

}