How can i display pagination in custom comment list?

You should use append paginate_comments_links() into $output like so:

add_shortcode ( 'show_recent_comments', 'show_recent_comments_handler' );

function show_recent_comments_handler( $atts, $content = null )
{
    extract( shortcode_atts( array( 
        "count" => 20,
        "pretty_permalink" => 0
        ), $atts ));

    $output=""; // this holds the output

    if ( is_user_logged_in() )
    {
    global $current_user;
    get_currentuserinfo();

    $args = array(
        'user_id' => $current_user->ID,
        'number' => $count, // how many comments to retrieve
        'status' => 'approve'
        );
    $comments = get_comments( $args );
    if ( $comments )
    {
        $output.= "<ul>\n";
        foreach ( $comments as $c )
        {
        $output.= '<li>';
        if ( $pretty_permalink ) // uses a lot more queries (not recommended)
            $output.= '<a href="'.get_comment_link( $c->comment_ID ).'">';
        else
            $output.= '<span id="cm-txt">متن دیدگاه : </span><a href="'.get_settings('siteurl').'/?p='.$c->comment_post_ID.'#comment-'.$c->comment_ID.'">';         
        $output.= wp_trim_words($c->comment_content, 20, '...');
        $output.= '</a><p>در پست: ' . get_the_title($c->comment_post_ID);
        $output.= "</li>\n";
        }
        $output.= '</ul>';
        $output .= paginate_comments_links(['echo' => false]); // echo false means that it would return the code instead of echoing it
        }
    }
    return $output;
}