Adding subdomain to home_url for “add_menu_page”

The way of your approach is arguably wrong. One shouldn’t use add_menu_page for external links as that function isn’t created for that purpose. Add menu page adds a page in the admin area with menu_slug as the fourth argument.

If we take a look at add_menu_page/dev ref source, we can notice that whatever the URL we pass in the function is passed through plugin_basename / dev ref which normalizes and replaces the path giving us the path relative to plugin directory.

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Instead you can hook it to admin bar which in my opinio correct place to put external link(if this is only purpose)

function wpse243214_user_manual($wp_admin_bar) {
    $wp_admin_bar->add_menu( array(
        //'parent'    => 'wpse-parent-id', // parent if any
        'id'        => 'wpse243214_usermanual',
        'title'     => __('User Manual'),
        'href'      => 'http://docs.'.basename(esc_url(home_url())),
    ) );
}
//change priority(15) accordingly
add_action( 'admin_bar_menu', 'wpse243214_user_manual', 15 );

With this we get

Also note that I used basename for home_url otherwise we get a link with http or https scheme (http://www.example.com).