How can hide widget title if custom widget is empty

I always move my filters inside my query, this will inhibit them from showing anything when my query returns no results. You just need to move the following

echo $args[ 'before_widget' ];      

if ( !empty( $title ) ) 
   echo $args['before_title'] . $title . $args['after_title']; 

to just after this line

if ( $related_posts ) { ?>

EDIT

I have quickly rewritten one of my widgets to accommodate my related posts function. I have commented a few places where there was issues. Hope this help

<?php
/**
 * Related_Posts_Widget widget class
 *
 * Displays posts from a selected category
 *
 * @since 1.0.0
*/
class Related_Posts_Widget extends WP_Widget 
{

    public function __construct() 
    {
        parent::__construct(
            'widget_related_posts', 
            _x( 'Related Posts Widget', 'Related Posts Widget' ), 
            [ 'description' => __( 'Display a list of related posts.' ) ] 
        );
        $this->alt_option_name = 'widget_related_posts';

        add_action( 'save_post', [$this, 'flush_widget_cache'] );
        add_action( 'deleted_post', [$this, 'flush_widget_cache'] );
        add_action( 'switch_theme', [$this, 'flush_widget_cache'] );
    }

    public function widget( $args, $instance ) 
    {
        $cache = [];
        if ( ! $this->is_preview() ) {
            $cache = wp_cache_get( 'widget_rel_posts', 'widget' );
        }

        if ( ! is_array( $cache ) ) {
            $cache = [];
        }

        if ( ! isset( $args['widget_id'] ) ) {
            $args['widget_id'] = $this->id;
        }

        if ( isset( $cache[ $args['widget_id'] ] ) ) {
            echo $cache[ $args['widget_id'] ];
            return;
        }

        ob_start();

        $title          = ( ! empty( $instance['title'] ) ) ? $instance['title'] : __( 'Related Posts' );
        /** This filter is documented in wp-includes/default-widgets.php */
        $title          = apply_filters( 'widget_title', $title, $instance, $this->id_base );
        $number         = ( ! empty( $instance['number'] ) ) ? absint( $instance['number'] ) : 5;
        if ( ! $number ) {
            $number = 5;
        }
        $cat_id         = $instance['cat_id'];
        $thumbnail      = $instance['thumbnail'] ? true : false; 

        /**
         * Filter the arguments for the Related Posts Widget.
         *
         * @since 1.0.0
         *
         */
        $cat_select = [ 
            'numberposts' => $number,
            'cat'         => $cat,  
        ];

        if ( function_exists( 'get_related_posts' ) ) { // This line was wrong, get_related_posts_widget should be get_related_posts
            $related_posts = get_related_posts( 'authors', $cat_select ); 
                if ( $related_posts ) {

                    echo $args['before_widget'];
                    if ( $title ) { // This should not be !$title
                        echo $args['before_title'] . $title . $args['after_title'];
                    }               
                    ?>

                    <ul class="related-cat">
                        <?php foreach ( $related_posts as $post ) {
                            setup_postdata( $post ); ?>
                            <li>
                                <?php if ( $thumbnail ) : ?>
                                <figure class="post-thumbnail">
                                    <?php the_post_thumbnail( 'tie-large' ); ?>                                               
                                </figure>
                                <?php endif; ?>
                                <h3><a href="<?php echo get_the_permalink( $post->ID ); ?>"> <?php echo get_the_title( $post->ID );?> </a></h3>
                            </li>
                        <?php } //foreach ( $related_posts as $post ) ?>
                    </ul>


                    <?php 
                } // end if     

            wp_reset_postdata(); // You forgot this, this is very very very important

            echo $args['after_widget']; 
        }

        if ( ! $this->is_preview() ) {
            $cache[ $args['widget_id'] ] = ob_get_flush();
            wp_cache_set( 'widget_rel_posts', $cache, 'widget' );
        } else {
            ob_end_flush();
        }
    }

    public function update( $new_instance, $old_instance ) 
    {
        $instance                   = $old_instance;
        $instance['title']          = strip_tags( $new_instance['title'] );
        $instance['number']         = (int) $new_instance['number'];
        $instance['cat_id']         = (int) $new_instance['cat_id'];
        $instance['thumbnail']      = $new_instance['thumbnail'];
        $this->flush_widget_cache();

        $alloptions = wp_cache_get( 'alloptions', 'options' );
        if ( isset($alloptions['widget_related_posts']) )
            delete_option('widget_related_posts');

        return $instance;
    }

    public function flush_widget_cache() 
    {
        wp_cache_delete('widget_rel_posts', 'widget');
    }

    public function form( $instance ) 
    {

        $title      = isset( $instance['title'] ) ? esc_attr( $instance['title'] ) : '';
        $number     = isset( $instance['number'] ) ? absint( $instance['number'] ) : 5;
        $cat_id     = isset( $instance['cat_id'] ) ? absint( $instance['cat_id'] ) : 1;
        $thumbnail  = isset( $instance['thumbnail'] ) ? $instance['thumbnail'] : false; 
        ?>

        <p>
            <label for="<?php echo $this->get_field_id( 'title' ); ?>"><?php _e( 'Title:' ); ?></label>
            <input class="widefat" id="<?php echo $this->get_field_id( 'title' ); ?>" name="<?php echo $this->get_field_name( 'title' ); ?>" type="text" value="<?php echo $title; ?>" />
        </p>

        <p>
            <label for="<?php echo $this->get_field_id( 'number' ); ?>"><?php _e( 'Number of posts to show:' ); ?></label>
            <input id="<?php echo $this->get_field_id( 'number' ); ?>" name="<?php echo $this->get_field_name( 'number' ); ?>" type="text" value="<?php echo $number; ?>" size="3" />
        </p>

        <p>
            <label for="<?php echo $this->get_field_id('cat_id'); ?>"><?php _e( 'Category Name:' )?></label>
            <select id="<?php echo $this->get_field_id('cat_id'); ?>" name="<?php echo $this->get_field_name('cat_id'); ?>">
                <?php 
                $this->categories = get_categories();
                foreach ( $this->categories as $cat ) {
                    $selected = ( $cat->term_id == esc_attr( $cat_id ) ) ? ' selected = "selected" ' : '';
                    $option = '<option '.$selected .'value="' . $cat->term_id;
                    $option = $option .'">';
                    $option = $option .$cat->name;
                    $option = $option .'</option>';
                    echo $option;
                }
                ?>
            </select>
        </p>

        <p>
            <label for="<?php echo $this->get_field_id('thumbnail'); ?>"><?php _e( 'Hide post thumbnail' ); ?></label>
            <?php $checked = ( $thumbnail ) ? ' checked=\"checked\" ' : ''; ?>
            <input type="checkbox" id="<?php echo $this->get_field_id( 'thumbnail' ); ?>" name="<?php echo $this->get_field_name( 'thumbnail' ); ?>" value="true" <?php echo $checked; ?> />    
        </p>

    <?php
    }

}

add_action( 'widgets_init', function () 
{
    register_widget( 'Related_Posts_Widget' );
});