Was looking into the virtual debit cards in NZ and one thing I noticed is that unlike with privacy.com in the US (and I think some others) which are specifically promoted with the ability to be used effectively pseudonymously as you can use any name and billing address, this doesn't seem to be the case for options available in NZ. (Or EU for that matter.)
Most people in NZ seem to use Wise for virtual cards although they only allow 3 simultaneously. More significantly, the name and billing address on even virtual card cannot be changed and is actually your name and personal address on the Wise account. AFAICT, these cards are Visa debit cards so generally work even on sites and services which reject prepaid cards although I assume the service could reject Wise.com cards specifically if they wanted too. And if your account is registered in NZ, I'm fairly sure your card is technically a NZ card.
It seems Revolut also provides virtual cards in NZ, I believe you can create 4 multiuse every 30 days up too a maximum of 20 simultaneous; and 5 single use a day (only one can be active). Otherwise it seems it's similar to Wise, name and billing address can't be changed and are you actual name and billing address, Visa debit cards, and you get a NZ card. Since Revolut seems more popular in the EU for this purpose, I sort of expect they're more likely to be banned by merchants.
Klara also seems to suggest you can create virtual Visa cards in NZ, including single use ones, but I didn't check very well. I would assume besides differences in the numbers you can create, it's otherwise similar to Wise and Revolt.
(Airwallex may also offer virtual cards in NZ, but it's for businesses.)
The obvious limitation here for pseudonymous use is the name and billing address attached to the virtual card is technically your real name and billing address. However, I'm not sure if it matters. I believe NZ debit cards still don't support the Address Verification Service (AVS) so I guess can actually enter any billing address you want, it cannot be checked even if the actual billing address is your real one? I can't find any recent comments, but I'm fairly sure it used to be the case that AVS wasn't supported in NZ. AVS often only check the numerical details so another partial out, but Visa seems to suggest the first 20 characters may be checked (now?).
Likewise it seems Account Name Inquiry (ANI) is very new. So I strongly suspect NZ cards still don't support it.
So I think you can effectively use Wise and Revolut virtual cards pseudonymously if based in NZ since although the name and billing address on the card might technically be your real name and address, you don't have to provide your actual name and billing address unless required by NZ law. Possibly AVS (and ANI?) is supported in parts of the EU where discussion over this is generally focused (mostly for Revolt). Or is there something I'm missing?
To be clear, I'm only thinking about simple pseudonymous use on the merchant side where you don't have to provide them your actual details unless they verify it in some other way or you're legally required to. I'm aware YKC means it's quite difficult and potentially illegal for the card provider to not know who you are (which AFAIK applies even to privacy.com stuff). And I'm sure they will cooperate with law enforcement etc as needed. Likewise I'm sure a merchant could go to court or otherwise convince Wise, Revolut etc to give up your details in certain scenarios. And if you're having something shipped, you'll need to provide an address their shipping provider can deliver stuff to. None of that is what I'm wondering about.
Cheers
