get taxonomy thumbnail and use it as a variable in code

You have ‘taxonomy’ => ‘product_cat’ in your code, so most likely you have Woocommerce, and we need to get a category picture from there.
Try this code

<?php           $args = array(
                'taxonomy'   => 'product_cat',
                'title_li'   => '',
                'hide_empty' => 0,
                'echo'       => 0,
                'show_count' => 0,
                'style'      => '',
                'parent'    => 0
                );

                $get_cats = get_terms( $args );
?>
<div class="design">
   <div class="menu">
    3 menu options will be in this div and when we hover at each link an image must appear to the right with fast zoom in effect like shooting.
       <ul class="hoverimage">
<?php

                foreach ( $get_cats as $cat ) {

                $thumbnail_id = get_woocommerce_term_meta( $cat->term_id, 'thumbnail_id', true ); 
                $image = wp_get_attachment_url( $thumbnail_id ); 
    if($image == "") {
    $image = get_bloginfo('url') . '/wp-content/uploads/2019/06/Logo-02-02.jpg';
    }
            ?>

        <li class="menu-item" data-image="<?php echo $image; ?>"><a href="https://wordpress.stackexchange.com/questions/352219/<?php echo get_term_link( $cat ); ?>"><?php echo $cat->name; ?></a></li>

            <?php
                }
            ?>
       </ul>
   </div>
    <div class="zoom-preview">
image will appear here when we hover at each link at the left. each link it's own image.
        Also there must be a default image here when we do not hover at any link at the left.
        <div class="img_conatainer">
        </div>
    </div>
   <div class="clear"></div>
</div>

  <script src="https://code.jquery.com/jquery-3.3.1.min.js"
              integrity="sha256-FgpCb/KJQlLNfOu91ta32o/NMZxltwRo8QtmkMRdAu8="
              crossorigin="anonymous"></script>         
            <script>


$(function(){
    var imageContainer=".img_conatainer",
        imageList=".hoverimage",
        maxWidth="parent", // parent or specific css width/  
        defImage="http://cdn.instructables.com/FEB/Q2RO/H9G190RM/FEBQ2ROH9G190RM.LARGE.jpg";
    var $imageContainer = $(imageContainer).eq(0);
    var $imageList      = $(imageList).eq(0);
    if (maxWidth === 'parent') { maxWidth = $imageContainer.width() + 'px'; }
    //Load images and set hover::
    $imageList.find('li').each(function(index){
        if (typeof $(this).data('image') === 'string') {
            $imageContainer.append(
                "<img id='imageToggle"+index+
                "' src=""+$(this).data("image')+
                "' style="max-width: "+maxWidth+"; display:none;" />"
            );
            $(this).data("image","imageToggle"+index);
            $(this).hover(
                function(){ loadImage($(this).data('image')); },
                function(){ loadImage('imageToggleDef'); }
            );  
        }
    });
    //Load default:
    $imageContainer.append(
                "<img id='imageToggleDef"+
                "' src=""+defImage+
                "" style="max-width: "+maxWidth+"" />"
    );
    //Toggle images:
    function loadImage(imageId) {
        $imageContainer.stop(true).fadeOut('fast', function(){
            $(this).find('img').hide();
            $(this).find('img#'+imageId).show();
            $(this).fadeIn('fast');
        });
    }

});

            </script>