Australia and New Zealand have both free trade and free movement of people between them. They aren't formally linked – free trade is a treaty, free movement of people is just an "arrangement" (not a legally binding agreement under international law, rather a political agreement implemented in each country's immigration law) – but it is all part of the overall multifaceted bilateral relationship.
Well, that gets the heart of what's do revolutionary about the EU's basic premises. "You want to move capital and profits and materials without limit? Sure, no problem, but only if labor has the same freedom."
I think a big difference is that Australia and New Zealand are only two countries, so there has never been felt to be a need to establish many joint institutions. Issues can generally be worked through by one-on-one negotiation. Rather than some single overarching body to manage everything, there are lots of individual agreements and arrangements on separate issues.
There are some joint institutions – for example, Food Standards Australia New Zealand (FSANZ), which is in charge of food regulation for both countries – but rather than an international bureaucracy, it is an Australian government agency. Even its Wellington office, all the employees are technically Australian public servants, even though they are mostly New Zealanders. New Zealand pays the Australian government part of FSANZ's budget; FSANZ issues new food regulations under Australian law and then the New Zealand government copies/pastes them into New Zealand law too: with rare exceptions, they are allowed to refuse to adopt a change if they disagree with it–Kava is a recent example, Australia has been cracking down on Kava due to concerns about the health risk, and using both controlled substances and food standards regulation to do so; New Zealand has been unwilling to go along due to its cultural importance to New Zealand's Pasifika (Pacific Islander) minority.
The issue is that the neighbouring countries are generally a lot poorer, and many of them are small island states, often with a shaky track record on democracy, so it just isn't politically and economically feasible to integrate them into the same economic relations as Australia and New Zealand have. There has been some talk of turning the Pacific Islands Forum, the regional international organisation, into a "Pacific Union", by expanding Australia/New Zealand's de facto economic union to the whole bloc. But realistically I think that's decades away, if it ever happens. And unless and until that happens, there is no real need to put the Australia/New Zealand economic union on a more formal footing.
