Welcome to the civil law world, where they don't need to prove that at all, they just need to ask "on the balance of probabilities, is it more likely than not that a violation happened here?". Unfortunately, for many, juries, and even judges, the mere assertion thereof by a corporation, even if backed by scant or contradictory evidence, is enough for them to say, "that's reasonable".
The rightsholder does not want to fight you in court because the amount of money you can pay (most likely) is less than what they have to pay in legal fees to hire these guys to come after you.
The era of mass lawsuits ended like 10+ years ago. It was a hallmark of the 2000's. These days, rightsholders do the settlement fishing where they get enough data to subpoena your identity from your ISP, then send you settlement offers. They don't want you to show up in court, they want to spook you and get you to give them $1000 or something to make it go away.
You legally purchased the music and legally made backup copies of it. You are not a criminal.
> I can't prove I own my own music if challenged
I don't think that matters. Someone would need to prove you stole it, which they can't do since you didn't.