>Isn't it possible that they automatically send the claims, but manually go over the counter claims? //
That's entirely possible but the company making the claim that they owned copyright is committing a [serious IMO] fraud and should be punished severely for it. If copyright infringement is a such a crime as the corporate world tries to suggest then false claim on copyright is far worse. The company knows that they will get false positives but continues.
One reason it's worse than mere infringement is it's an attempt to deprive either the public domain or the original owner in order to make financial gains whilst mere infringement is usually just for use. It's much closer to theft than plain infringement as it seeks to deprive the owner (or public domain) of the enjoyment (i.e. use or licensing) of the work whilst mere infringement generally does not.
AFAIK such fraudulent claims to ownership are not treated at all harshly in IP law? This is a case where it should be considered criminal [rather than just as a tort] as generally it's the action of a larger organisation against a content creator - the state should protect its citizens particularly when there are such imbalances of power.