There is no preemptive protection for this. Comments matching certain patterns are automatically deleted when you flag them (not sure where that is formally documented) but "thanks" isn't one of the patterns. Currently, you have to take action after the fact.
So, right now, you can:
- Flag them as "too chatty" (if you mouse over the comment a little flag will appear on the left that you can click), or
- Flag the entire post with a custom flag asking for a comment cleanup if there are a ton of them.
Then, a mod will eventually handle it.
I've gone through and flagged the comments on that post (the flags were handled within a couple hours and the comments are now cleaned up). That post's comments were... unusually bad.
Comment protection similar to post protection (raised rep threshold) would be problematic as you don't want to prevent new users from leaving relevant comments. The only reasonable option, if protection were added, would be to e.g. add short comments containing "thanks" to the auto-delete-on-flag filter, which is a good balance of easy deletion and required human intervention (still, it would be difficult to create an accurate filter that also matched everything in your example). On the other hand, the current flagging mechanism is effective and relatively quick.