This is what happens when corporations leverage their size to implement systems that bypass established Law, Copyright & DMCA in this instance. Publishers, like BMG, in wanting a quicker way to effectively take down content without doing through legal channels (aka DMCA), push for and abuse systems like Content ID. Claimants (copyright 'owners') are then not technically breaking the Law, because these types of systems are 'internal' and based on the voluntary policing of content
based on Terms of Service, rather than Copyright, violations. In YouTubes case for example, when a Vlogger uploads they do so stating implicitly that they own the content and all rights therein. When a claim comes in countering this, YouTube can claim a ToS violation to be dealt with appropriately as is their want.
This is why Content ID is dangerous; it allows publishers ('Claimants') to make
'soft' copyright claims against users that typically result in take-downs, disablement's, or revenue share/confiscations without specifically having to invoke full Copyright claims and remedy through DMCA (or it's localised equivalents), which would broadside them for prosecution where a claim would have to be explicitly clarified.
Content ID is what happens when corporations abuse their power to find ways around established Law. And make no mistake, they will have had plenty of dark, smoky room conversations with lawyers to make sure none of them can be held liable for anything as a result of Creators incomes being screwed with. So be warned.. if music in particular does need to be used, do your due-diligence, and cite sources (give credit) where necessary to avoid, if not mitigate, the types of mess the chap linked to above found himself in.
Incidentally, just because the song itself may be in Public Domain, that should not be taken to mean the manuscript, arrangement or other aspect of the music used is. And that's where the trap lies. So be warned.