How can I catch WordPress custom settings page slug has already changed?

you can use the admin_init hook along with the add_query_arg() function to modify the redirection URL for handle the situation.

Redirect users to the correct URL admin.php?page=management if they access the settings page with an unexpected slug.

Modify the redirection URL after settings have been updated to include the current page slug management, ensuring that users have redirected back to the correct page admin.php?page=management&settings-updated=true

add_action('admin_init', 'check_custom_settings_page_redirect');

function check_custom_settings_page_redirect() {
    // Check if we are on the settings page and it's not the expected slug
    if (isset($_GET['page']) && $_GET['page'] !== 'management') {
        // Redirect to the correct URL with the updated slug
        $redirect_url = admin_url('admin.php?page=management');
        wp_safe_redirect($redirect_url);
        exit;
    }

    // Check if settings have been updated
    if (isset($_GET['settings-updated']) && $_GET['settings-updated'] === 'true') {
        // Modify the redirect URL to include the current page slug
        $redirect_url = add_query_arg(array('page' => 'management'), admin_url('admin.php'));
        wp_safe_redirect($redirect_url);
        exit;
    }
}