In Australia they have to record the details of the person buying the phone, but suffice it to say that relying on someone making minimum wage at a supermarket to enforce national security isn't particularly effective.
Just buying a phone at the supermarket doesn't make the SIM work.
When I registered my last SIM, I did it online and they wanted Drivers Licence number and a lot of other info that I felt was quite a lot just so I could use a phone.
Yeah they try to track when people have lots of phones registered to them. But last time I did it they were not able to correctly validate international passport numbers. Or you could just buy validated sim cards from poor people or international travellers.
Which provider? I noticed (too late) that the airport SIMs have special overpriced plans. I'm used to paying ~$10/GB for data in Australia, it seems to be about double that in NZ without going on a plan.
I bought a 2 degrees one, but I'm pretty sure they had Skinny and Spark as well at the local New World. Easy to setup with just a credit card, or could've used one of the voucher cards which I'm pretty sure can be bought with cash.
Both here and in the US, I've found that airports usually are not good places to get SIMs - there seems to be some agreement whereby the expensive ones aimed at tourists are all that's available. Supermarkets or department stores tend to be my go-to.