Products like this are particularly susceptible to fraud, especially given that they're charging your bank account directly. Without any KYC, if I as a fraudster get access to your bank login, I can basically drain your account. That opens up a ton of liability to Privacy. Also, the card networks tends to be pretty risk-averse with allowing issuing products access to their networks if there's a lot of fraud going through their services.
So, it's much easier to put KYC in place in order to avoid the headache of dealing with fraud/AML/etc
The actual funding source/bank behind them won't see any detail beyond "privacy.com".
Any card issuer (or virtual card issuer) has to do this.
Edit: A sibling explains a good liability reason for this, which makes sense but would not imply any legal reason.