How do I check to see if a specific child term has shared posts with another set of child terms within the same taxonomy?

You’re definitely getting an array of what I believe is all of the models, but you should be able to use has_term like you did, but with the $model variable as what to check for. $model should be the ID number for the categories that you’re seeing.

And then just use get_cat_name by passing in $model. I’m at a loss if this doesn’t work.

// Get ready to query all posts under the Models term
  $args = array(
     'posts_per_page' => -1,
     'orderby' => 'ASC',
     'post_type' => 'download',
     'download_category' => 'models',
     'post_status' => 'publish'
  );

  // Query 'dem posts!
  $all_models_posts = new WP_Query( $args );


  if ( $all_models_posts->have_posts() ) {
    while (have_posts()){
      the_post();

      // Get all models
      $models = get_term_children(204, 'download_category');

      foreach($models as $model) {
           if(has_term($model, 'download_category')){
                echo get_cat_name($model);
        }
      }
    }
  }

  // Reset that query
  wp_reset_query();
 ?>