How do I add the “orange counter” next to a Custom Post Type?

I believe that the only sensible place to rename a menu item is the add_menu_classes filter hook.

It has a different destiny, but I can’t see any other place.
Hook passes an array of first-level menu items as a parameter to the function.
The array elements look like this:

$menu = [
    5 => Array (
        [0] => 'Menu title'
        [1] => 'capabilities'
        [2] => 'menu_slug' // e.g. edit.php | edit.php?post_type=page | edit.php?post_type=cpt_slug 
        [3] => 'Page title'
        [4] => 'css classes'
        [5] => ??
        [6] => 'dash-icon'
    ),
    70 => Array (
        [0] => 'Users'
        [1] => 'list_users'
        [2] => 'users.php'
        [3] => ''
        [4] => 'menu-top menu-icon-users'
        [5] => 'menu-users'
        [6] => 'dashicons-admin-users'
    ),
]

You can add badge with number by editing the menu item title, which is at index [0].

add_filter( 'add_menu_classes', 'se414500_menu_classes' );

function se414500_menu_classes ($menu)
{
    // --- v1 ---
    $my_counter = count( $something );
    foreach($menu as &$item)
    {
        // use menu_slug of your item
        if ($item[2] != 'edit.php?post_type=my_cpt_slug')
            continue;

        $item[0] .= sprintf(
            '<span class="update-plugins count-%s"><span class="update-count">%s</span></span>',
            $my_counter,
            number_format_i18n( $my_counter )
        );
    }
    // --- v2 ---
    // foreach($menu as &$item)
    // {
    //     // use menu_slug of your item
    //     if ($item[2] == 'edit.php?post_type=my_cpt_slug') {
    //         //
    //         // get the value to display next to the name
    //         $my_counter = count( $something );
    //
    //         $item[0] .= sprintf(
    //             '<span class="update-plugins count-%s"><span class="update-count">%s</span></span>',
    //             $my_counter,
    //             number_format_i18n( $my_counter )
    //         );
    //     }
    // }

    return $menu;
}

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