Add Plugin options as subpage to Theme options page

I have my own Theme with an options page. Now I am developing a new Plugin which should have an options page that should be a subpage of the Theme options page.

Theme options.php:

?php
function xx_theme_admin_menu() {
    add_menu_page(
        'Theme Settings',
        'Theme Settings',
        'manage_options',
        'xx-settings',
        'xx_theme_admin_page' ,
        '/wp-content/themes/xx/assets/img/logo-Grayscale-30x25.png'
    );
}
add_action( 'admin_menu', 'xx_theme_admin_menu' );

function xx_theme_admin_page() {
    ...
}

Plugin options.php:

?php
function xx_cookie_banner_admin_menu() {
    add_submenu_page(
        'xx-settings',
        'Cookie Banner',
        'Cookie Banner',
        'manage_options',
        'xx-cookie-banner',
        'xx_cookie_banner_admin_page'
    );
}
add_action( 'admin_menu', 'xx_cookie_banner_admin_menu' );

function xx_cookie_banner_admin_page() {
    ...
} ?

With the above code, as soon as I access the subpage, I see the error "You are not allowed to access this page". Also, I am getting this error for the Theme options page because it does now link to the subpage, too:

So both, Theme options page and Plugin options subpage have the link to the Plugin options subpage.

Without the Plugin options subpage, the Theme options page works perfectly.

Any ideas on this?