I got one of these earlier in the year and had the same question at the time...
It turns out that a guest card purchase does not show in the app, nor is there a visible electronic record when logged into an air NZ account in a browser (no idea why they wouldn't introduce this feature, perhaps it's not popular enough or maybe just not a priority of any sort?)
Anyway, after speaking to the call centre, I was told that only limited lounges have the capability to check for the existence of a guest card electronically. I don't know how true this statement is in reality, considering the card isn't even reflected in the online account.
The nice woman on the phone offered to instead send me a letter via email that I could show to prove I was a card holder at the lounge entrance. A letter which I never received (and I didn't bother to follow up on)
Fortunately I received the guest card in the mail a day before travel, so got my extra guest in no problem. It took approximately 7 days for this to arrive in my letterbox.
However, what they don't tell you is that your extra guest card is no good at a regional lounge if you're departing on a domestic jet. Found that bit a little confusing, especially so since my first guest was allowed in no problem (for those interested, that happened in Wellington, but before the lounge upgrade began)