- Most YouTube takedowns have nothing to do with the DMCA as YouTube has their own proprietary system. This system is extrajudicial. This system has the type of problems you're raising and little recourse.
I actually think the DMCA takedown process itself is problematic, but YouTube's own private one is a whole extra level of abusive.
People need to remember that laws reforming the DMCA would do nothing to fix YouTube's system, unless they outright banned YouTube/Google from operating an extrajudicial system.
Only in a very narrow sense of "false": if the issuer of the takedown claims they have copyright over something they don't. The statement that they have a copyright on some work is made under penalty of perjury.
The statement that your work infringes on their copyright is not made under any penalties at all, and there are no consequences for falsely claiming infringement.
So if you play the Moonlight Sonata, I can't claim that I own the copyright to the Moonlight Sonata and you're infringing. I can claim that I own the copyright to some actual work I own the copyright to (even if totally unrelated) and that your work infringes it.
You could try to bring an action joining youtube but google is so wealthy that it's nearly essentially immune to litigation, including by states-- so it would be a waste of time.
If I post a video of myself playing moonlight sonata to youtube, and somebody puts a copyright claim (or more than one) against it, then it gets taken down and I suffer the consequences, but there is no mechanism for penalties for what amounts to fraud by the claimant. They should automatically get fined (payed to both me and youtube), and if damages were more severe I should be able to bring them to court to receive larger damages. Claimants should also get strikes the same way posters due, too many false claims and you then have to manually prove the copyright is yours before youtube takes the video down.
The above example was for youtube, but broadly this is how it should work. If copyright protection action cause harm, because of false claims (either accidental or intentional) then the claimant gets punished.