WordPress add menu items display slug too

To directly answer your question, you can use the wp_setup_nav_menu_item filter to change properties of the nav menu item:

add_filter( 'wp_setup_nav_menu_item', static function ( $menu_item ) {
    if ( ! is_admin() || empty( $menu_item->taxonomy ) || 'category' !== $menu_item->taxonomy ) {
        return $menu_item;
    }

    $menu_item->title .= sprintf( ' (%s)', $menu_item->slug );

    return $menu_item;
} );

The above approach may have unintended consequences, and pollutes the list of terms. A cleaner approach would be to intercept the AJAX call and implement your own search logic:

add_action( 'wp_ajax_menu-quick-search', static function () {
    if ( ! current_user_can( 'edit_theme_options' ) ) {
        wp_die( -1 );
    }

    require_once ABSPATH . 'wp-admin/includes/nav-menu.php';

    $request = $_POST;

    $args            = array();
    $type            = isset( $request['type'] ) ? $request['type'] : '';
    $object_type     = isset( $request['object_type'] ) ? $request['object_type'] : '';
    $query           = isset( $request['q'] ) ? $request['q'] : '';
    $response_format = isset( $request['response-format'] ) ? $request['response-format'] : '';

    if ( ! $response_format || ! in_array( $response_format, array( 'json', 'markup' ), true ) ) {
        $response_format="json";
    }

    if ( 'markup' === $response_format ) {
        $args['walker'] = new Walker_Nav_Menu_Checklist();
    }

    if ( ! preg_match( '/quick-search-(posttype|taxonomy)-([a-zA-Z_-]*\b)/', $type, $matches ) ) {
        _wp_ajax_menu_quick_search( $_POST );
        wp_die();
    }

    if ( 'taxonomy' !== $matches[1] ) {
        _wp_ajax_menu_quick_search( $_POST );
        wp_die();
    }

    // First, let's try to match our query with term's slug.
    $terms = get_terms(
        array(
            'taxonomy'   => $matches[2],
            'search'     => $query,
            'number'     => 10,
            'hide_empty' => false,
        )
    );

    // First search failed. Default to core logic: search by name.
    if ( empty( $terms ) || is_wp_error( $terms ) ) {
        $terms = get_terms(
            array(
                'taxonomy'   => $matches[2],
                'name__like' => $query,
                'number'     => 10,
                'hide_empty' => false,
            )
        );
    }

    if ( empty( $terms ) || is_wp_error( $terms ) ) {
        wp_die();
    }

    foreach ( (array) $terms as $term ) {
        if ( 'markup' === $response_format ) {
            echo walk_nav_menu_tree( array_map( 'wp_setup_nav_menu_item', array( $term ) ), 0, (object) $args );
        } elseif ( 'json' === $response_format ) {
            echo wp_json_encode(
                array(
                    'ID'         => $term->term_id,
                    'post_title' => $term->name,
                    'post_type'  => $matches[2],
                )
            );
            echo "\n";
        }
    }

    wp_die();
}, 0 );

For this approach, it copies a lot of code from WP core, so there may be incompatibilities in future versions.