If Google Analytics module is installed and it has a valid account ID, it would work on any other page except in share-this-thing-html.tpl.php

I tried changing JavaScript scope, or locally caching tracking code file in GA's Advanced Settings. But it didn't change anything.

If GA is configured to use "footer" as the JavaScript scope, it wouldn't work, because Share This Thing doesn't have a $footer region, this is expected.
However, "header" scope should work for GA as well, just like all other JavaScript files do.

Until this issue is resolved, here is a temporary solution:

The "googleanalytics_account" variable's value is passed to share-this-thing-html.tpl.php via template_preprocess_share_this_thing_html().

Copy share-this-thing-html.tpl.php to your theme folder.
Paste following code before closing body tag:

<?php if ($googleanalytics): ?>
<script>
  var _gaq=[['_setAccount','<?php print $googleanalytics; ?>'],['_trackPageview']];
  (function(d,t){var g=d.createElement(t),s=d.getElementsByTagName(t)[0];
  g.src=('https:'==location.protocol?'//ssl':'//www')+'.google-analytics.com/ga.js';
  s.parentNode.insertBefore(g,s)}(document,'script'));
</script>
<?php endif; ?>