Replace “WordPress” word in title of Dashboard

You’ve tried correct filter – just needs to update return:

function my_admin_title ( $admin_title, $title ) {
    return $title . ' ‹ ' . get_bloginfo( 'name' ) . ' — ' . 'foobar';
}
add_filter( 'admin_title', 'my_admin_title', 10, 2 );

Btw, filter above works only for logged pages, for login page needs to add another filter:

add_filter( 'login_title', 'my_admin_title', 10, 2 );