As far as I'm aware, the 8th Amendment does not apply here. I mean, just because the punishment is high doesn't mean it's cruel, and it's certainly not unusual, if it applies to everyone. There are certainly some defendants who would not be deterred without the ability of copyright law to generate absurdly high damage awards. And, of course, nothing legally stops a court from assigning punitive damages in a copyright case. In fact, that probably would have worked out better for the Internet age than the law the MAFIAA[1] bought.

That being said, a lot of constitutional provisions don't apply the moment you step foot in a civil court. For example, you actually can be compelled to self-incriminate, 4th Amendment be damned[0], so long as it's not a criminal proceeding. Likewise, there's caselaw stating that the 8th Amendment flat-out does not apply until the US is named as a party on the lawsuit.

On the other hand, SCOTUS has also thrown out punitive damage awards on 14th Amendment due-process clause grounds. In this case[2], we even have a math formula: punitive damages cannot exceed 10x the compensatory damages. Of course, because copyright already has very high statutory damages, we rarely even need to impute punitive damages to get billion dollar awards.

This is all dancing around another question, though: why do the damages have to fit the crime and not the person? Europe assigns scaling damages based on the defendant's ability to pay, and that would neatly solve the problem of well-pocketed copyright scofflaws that Congress attempted to fix with a sledgehammer. The problem is, American law doesn't actually do this. As far as I'm aware, it's not outright unconstitutional to scale fines to income, but given that it's unusual, I could imagine SCOTUS also finding it to be cruel. I mean, you are singling out the rich for being rich, and America was built to protect the interests of the rich.

[0] When I asked Gemma 4 what it thought of an earlier version of this post, it pedantically pointed out that the only legal compulsion civil courts can apply is an adverse inference - i.e. juries and judges in civil court are allowed to assume you're hiding evidence of guilt, whereas in criminal court they're not. I don't think this distinction matters.

[1] RIAA + MPAA = ???

[2] https://supreme.justia.com/cases/federal/us/538/408/