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.
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
What this kind of article should be about is the same thing that VPN is about. privacy.com actually removes privacy. Now, instead of VISA being the only aggregator of your purchase data, privacy.com is also. Either way, the merchant and VISA know your purchase history. Now, so does "privacy".com.
If you want to hide something from your bill so that someone else in your household doesn't see it, then ok. If you want to hide your address and name from a merchant (say for a digital downlaod) then ok. If you actually want more privacy, privacy.com is the opposite of what you want.
how does the merchant know your purchase history if you're using virtual cards with bogus info?