You can try using a custom function hooked in woocommerce_cart_item_name
filter hook, this way:
add_filter( 'woocommerce_cart_item_name', 'shipping_class_in_item_name', 20, 3);
function shipping_class_in_item_name( $item_name, $cart_item, $cart_item_key ) {
// Only in cart page (remove the line below to allow the display in checkout too)
if( ! ( is_cart() || is_checkout() ) ) return $item_name;
$product = $cart_item['data']; // Get the WC_Product object instance
$shipping_class_id = $product->get_shipping_class_id(); // Shipping class ID
$shipping_class_term = get_term( $shipping_class_id, 'product_shipping_class' );
if( empty( $shipping_class_id ) )
return $item_name; // Return default product title (in case of)
$label = __( 'Shipping class', 'woocommerce' );
return $item_name . '<br>
<p class="item-shipping_class" style="margin:12px 0 0;">
<strong>' .$label . ': </strong>' . $shipping_class_term->name . '</p>';
}
Add this to your functions.php file of the active theme. Tested on Woocommerce 3.5.