IMO, I think the approaches are similar - especially when compared with other countries around the world. At the moment NZ is in what they call Alert Level 2 (with the exception of Auckland in Alert Level 4).
Here's a quote from it: Ms Ardern said stamping out Covid-19 remains the best approach for New Zealand “for now,” but the goal is to get past using lockdowns and a border closure as the primary defences.
“Our collective goal is to move away from having to use these measures in that way and vaccines help us do that. No one wants to use lockdowns forever and I can tell you now that is not our intention,” she said. “But, for now, while we vaccinate, elimination is the goal.”
Often in Australia you'll hear the term "supression" instead of "elimination" but I think this semantic difference is politically motivated more than anything else - we're effective reacting very similar. IMO, we do take an elimination strategy as we apply lockdowns and restrictions swiftly when there is community transmission of Covid. Obviously there are examples here where the restrictions weren't applied swiftly but compared to other country's repsonses it is swift.