Check which registered sidebar a widget is added to

There isn’t a standard way to do it AFAIK in the form() method; here’s a function to do it:

function wp158055_get_sidebar_id( $widget ) {
    foreach ( wp_get_sidebars_widgets() as $sidebar_id => $widget_ids ) {
        if ( array_search( $widget->id, $widget_ids ) !== false ) return $sidebar_id;
    }
    return false;
}

And call with $this. Note from the widget() method you can use $args['id'] as mentioned in this answer.