Try this :
$menu = get_term( $locations[$theme_location], 'nav_menu' );
$menu_items = wp_get_nav_menu_items($menu->term_id);
foreach( $menu_items as $menu_item ) {
$link = $menu_item->url;
$title = $menu_item->title;
$slug = basename($menu_item->url);
}
Pass the whole permalink to the basename
function, which will automatically process the URL and give us only the slug.