That has been my interpretation of the DMCA notices I've received in the past, its an issue between the host of the content and whoever sent the DMCA notice.
Under Section 512 the only responsibility of the service provider is to remove the content, relay the notice to the actual copyright infringer, and allow them to file a counter-notice.
From there if the counter-notice isn't accepted by the copyright owner, they would have to file a lawsuit with the copyright infringer.
But can you as the alleged infringer sue the claimant?
I think the original question is if the platform can sue the person sending the DMCA takedown, for some kind of harassment. Because the platform itself will also investigate before taking content down.
Essentially the belief is that DMCA would never be abused for any reason so you need to penalize the platform but not the submitter.