NZ and Australia have very different fauna. They might look close on a world map but the distance from Sydney to Auckland is over 2000 km.

Just one example: Australia has a wide variety of mammals (mostly marsupials, e.g. kangaroos). NZ has very few native mammals, but it does have a huge assortment of unusual native birds. (Indeed, birds fill a lot of ecological niches that mammals normally do.) In fact, there's an ongoing effort to get rid of all non-native mammals: http://www.newyorker.com/magazine/2014/12/22/big-kill

Finally, the "all the Animals in Australia want to kill you" idea is popular and kind of amusing, but hugely overblown. Yes, there are deadly animals but you typically have to be doing something unusual or silly to encounter one in a dangerous way. Plus there's no poison ivy...