Is anyone posting in this thread a NZ Asian?
It's kinda inappropriate for people who aren't to be saying that she shouldn't have been offended by it when they haven't experienced what it's like being an Asian, especially a Kiwi Asian, in New Zealand.
Racism is a more objective standard, and agree that there is likely no racism at play here unless the chefs or serving staff etc discriminated against her because she was described as "Asian" on the docket and I don't think any racism was intended - but it could be considered a form of casual racism, the sort many people brush off as ok, but over time it builds up and grates at a person until they break which may have happened here, even though this isn't one of the more egregious situations by far.
But as to whether something is offensive or not should be considered from the perspective of the person who may be feeling the offence. Telling someone they shouldn't be offended by something suggests that you think your opinion on how they should be feeling trumps theirs and you don't what they think or their feelings.
I can see why she would feel offended by it - the term Asian has been used in the past in offensive ways, when people refer to Asians as a group it is not normally in a nice way (or at least, those memories stay with you and are remembered easier) and being categorised as Asian presents a form of "otherness", like you don't belong in the country. It's the sort of thing white NZers who immigrated here don't experience. It's similar to how lots of Kiwi Asian people don't like the question "Where are you from?" and the worse "no, where are you really from?". It makes you feel like an outsider despite the fact that many Kiwi Asians have been here for generations. Kiwi Asians often have little connection to their ancestral homeland apart from their appearance but they get automatically categorised because of it.
Just because in many Asian countries there is a lot of casual racism shouldn't mean that we should tolerate it in NZ and that Asians in NZ should tolerate it just because white people tolerate it in Asia.