Question edited
How can I add a class to the body when the menu at the menu setting > theme locations is selected.
Like in this screenshot:
And if I have two menus, can it add two different classes to the body tag?
I would like to use that class to control the nav toggle icon, etc.
Use the has_nav_menu to detect if the menu is configure in a body_class filter.
Something like
add_filter( 'body_class', 'wpse209469_body_class' );
function wpse209469_body_class( $classes ) {
if (has_nav_menu('primary-menu'))
$classes[] = 'primary-menu';
return $classes;
}
If you have two menus you can change the condition to match both, or just do two filters. There is probably a way to enumerate all the menus and add classes for them but it is probably an overkill.
primary-menu. So just use the same function add a class to the body.