Your audience is relishing the unique experience they're having (the privilege of speaking to someone from the other side of the planet). I hope you try to take it as a compliment in the future rather than chastise them with caricatured accents to "get your point across". :(
I get that it gets old, though, as your accent isn't novel to you.
I would imagine that part of the reason it gets old is that it isn't "your accent." It is a NZ accent after having been filtered and ruined by US ears and speech patterns.
Old joke. What do you call someone who knows three languages? Trilingual. Two languages? Bilingual. One language? American (US). Speaking as one myself...
"Fush and chups" - New Zealand. "Feesh and cheeps" - Australian
If you want to tell Aussies and kiwis apart, listen for "Chance" and "Dance".
Kiwis will say the 'a' like in the word "aren't", whereas Aussies will say it like the 'a' in "ants".
Much more discrete to ask about dancing than fish and chips ;)
That said, both AU and NZ have their own "TV accents", which in AU is referred to as "Cultivated Australian", which is relatively close to British RP (though not the same), and I expect NZ has their own equivalent, but most people would have something close to what you would generally think of people from that part of the world sounding like.
Not the US south.
In terms of US-UK parallels, there's also the effect that geographical isolation has in making a sort of language "time capsule" where colonies stick with dated pronunciations or vocabulary. I've heard an attempted reconstruction of London accents from the early 1700s that sound a lot like General American or Canadian English. The same source had an early 1800s reconstruction that sounded rather Australian to my ears. One imagines that English people colonize either place at a given time, leave some trace of their time and place in speech patterns, and their relatives in England go on to evolve speech patterns in different ways.
Non rhotacism (lack of pronunciation of R) also left some traces in the American south (and northeast). That feature began in southern England after colonization, so I believe it was generally fashionable people keeping up with the latest English trends who brought it west.
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.
Of course, just the same as British. "American accent" for foreigners usually means "Southern Foghorn Leghorn" e.g. Daniel Craig in Knives Out, but can also mean New York, Boston, Chicago, Dakota (aka Fargo) or some other variation. To Americans it really means "unaccented" Midwestern English, aka broadcast (radio or TV) voice.
Americans know "American accent" from "British accent" by the same standard, the British "unaccented" voice is called RP and sounds posh or overly fancy to most Brits, but Americans don't distinguish RP from Cockney, Yorkie, Jordie or any of the other variants. RP is the broadcast voice.
To be clear: I used to be the same. "British" was one accent that Michael Caine, the Queen and Chris Ramsay all spoke. (For Americans: "Robert DeNiro, Harrison Ford and Dolly Parton have one accent" is equally wrong and hilarious.)
Seems pretty common.
Fortunately I can do a bit of California surfer dude right back and that usually gets the message across