Yes that is the telegraphic code:
see Origins of the acronym ANZAC | The Australian War Memorial
However one of the ways that Anzac is now also used is to describe the special ongoing relationship between Australia and New Zealand forged from that original joint service, and especially in times of trouble. Hence Anzac Bubble.
This relationship is like that of two brothers, a brotherly rivalry where we seem to 'give each other crap' about everything but where you always have each others back if something goes wrong.