It would be useful to know how much people are using comment thanking after it is launched.

There is no instrumentation involved with the DiscussionTools overflow menu or usage of the thanks tool within it. However, we can easily track thanks that came from the tool in the database because they have a distinctive identifier format:

$uniqueId = "discussiontools-{$comment->getId()}";

As such, I think that approximately this query would work:

SELECT COUNT(*) FROM logging WHERE log_type='thanks' AND log_action='thank' AND log_id LIKE 'discussiontools-%'

...with whatever elaborations we want around time ranges (log_timestamp), users (log_actor), and namespaces (log_namespace).