How to get update_post_meta output to include string length

Your data is being serialize properly. Your code saves an array of integers. Integers are not strings and so a string length is not saved. Run the following code and it should illustrate the point:

$str="a:4:{i:0;s:4:"7613";i:1;s:4:"7615";i:2;s:4:"7616";i:3;s:4:"7618";}";
$str = maybe_unserialize($str);
var_dump($str);
echo '<br/>';
var_dump(maybe_serialize($str));
echo '<br/>';
$str = array(
  1234,4567,8910,1112
);
var_dump(maybe_serialize($str));
echo '<br/>';
$str = array(
  '1234','4567','8910','1112'
);
var_dump(maybe_serialize($str));

To get the results you want, save strings and not integers:

// Fill the array with attachment ID's
foreach($attachments as $attachment) {
        $atts[] = "$attachment->ID";
}