Sure, you can pass on up to 4 arguments to that hook, you can find them in the linked documentation.
For example:
add_filter( 'asp_results', 'asp_custom_link_results', 10, 4 );
function asp_custom_link_results( $results, $search_id, $is_ajax, $args ) {
.....The $search_id variable holds the ID of the search shortcode.