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They don't have to convince any verifier. They don't have to be verified. If there's no official account and you create an account with a reasonable name, reposting every post from the official feed, you can get significant following. A lot of the followers will not care whether it's official or not and may not question an extra information appearing on the feed one day.

Well, put it this way: why is Donald Trump listed on Twitter as @realDonaldTrump?

If you don't snatch up your (organization's) name first, someone will surely do so for you.

(Honestly not trying to incite anything by using him as an example; I just hardly use Twitter and he was the first to come to mind.)

They own the non “real” one too, he’s just too much of a tool to use it.

Probably acquired later and didn't want to lose his followers.

Once a public official leaves office twitter should remove all followers to zero.