How to pass parameter that ends up being part of a class name with wordpress shortcode

The second argument of any WP shortcode is the content found inside the shortcode block. Additionally, Not sure how you are trying to use forward slash on a shortcode name, slashes are not allowed as a part of shortcode name. Try it like –

function colored_box_shortcode( $atts, $content )
{
   extract( shortcode_atts( array(
      'color' => 'grey',
   ), $atts));

  return 
  '<div class="shadow-wrapper half-shadow im-centered">
  <div class="box-shadow shadow-effect-2">
  <div class="servive-block servive-block-'. $color .'">'. $content .'</div>
  </div>
  </div>';
}
add_shortcode('box', 'colored_box_shortcode');