No, their accent is quite clearly identifiable as Kiwi to us Brits.

They're easy to understand, they enunciate when they speak but the intonation and inflections and general sound is very different.

As for "a British accent". Would an American say there's "an American accent"?

There is no single British accent, we have, English, Scottish, Welsh, Irish and others with their own languages (and thus accents), then in England, we have widely varying accents, many clearly identifiable to a Brit; Newcastle (Jordie), Birmingham (Brummie), Liverpool (Scouse), London (Cockney), Essex, Yorkshire, etc.

I think when people say "British" like the article in OP, they really mean what we call "posh". We do it ourselves at times, but most people don't actually speak that way.