Unable to get product category

You have a few things that are wrong.

remove global variable as we are already working in a custom loop no need for global $post or other globals.
Get all posts with that selected tag in the $args (assuming this is what you want).
If you know the type of request, I’d use $_GET or $_POST accordingly instead of $_REQUEST.
$terms is an object, so you need to iterate over it to get the info you want
Here I hooked this example to init but you might want to hook later depending on your code.

add_action('init', 'wpse_233955' );
function wpse_233955 (){
  if ( isset( $_REQUEST['prod_tag'] ) ) {
    $tags = $_REQUEST['prod_tag'];
 }

 $args = array(
    'post_type'      => 'product',
    'posts_per_page' => -1, // Get all post with the tag not just 1
    'product_tag'    => $tags
  );

  $query           = new WP_Query( $args ); // $query is you custom loop
  $product_count   = $query->post_count;

  if ( $product_count > 0 ) :

    while ( $query->have_posts() ) :

      $query->the_post(); // This will set the $query object to the current post item
      $thePostID = get_the_ID(); 
      $terms = wp_get_object_terms( $thePostID, 'product_cat');

      foreach( $terms as $term ){
        echo $term->name;
      }

      // ...
    endwhile;
  endif;
}

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