Translation:I don't have either tea or herbal tea, but I have coffee.
I'm a professional editor. It's just wrong. The "either" does not belong there under any circumstance. It's a total misunderstanding of that word's meaning and usage. It also shows the writer is ignorant of the use of "neither . . . nor", which would be correct in that sentence.
As a professional you ought to know that is incorrect.
The two expressions are logically identical, although I agree with you that "neither … nor" is a superior way to express that logic, and as an editor it is the one that you should be insisting upon.
But you need to swot up on your references.
Try something like:
R. Huddleston & G.K. Pullum (ed.), The Cambridge Grammar of the English Language, Cambridge University Press 2016, p. 1310.
All three of the following are logically equivalent: 
i She found it neither surprising nor alarming. (=[48i])
ii She didn’t find it either surprising or alarming.
iii She found it both not surprising and not alarming.
As someone who has spoken British English as his native tongue for over 70 years, I agree with GraemeSarg and Roody (both native speakers I believe). You will hear Duo's construction in spoken English ALL THE TIME and no-one thinks twice about it or calls the Grammar Police. If you use the 'neither ... nor' in everyday speech people are liable to think you are 'showing off' a bit. It is more likely to be found in written English, which may explain why a professional editor prefers it. But the construction is natural and grammatical and is found in respected publications such as 'The (London) Times, 'The Guardian' or 'The Economist'. (I will now get downvoted en masse so this will never be seen again!)
The construction used by Duo is correct.
"Whereas in English you have three ways to express the negation - not either… or / neither… nor… / not... or... - in French, you only use ne... ni... ni..."
I added the emphasis. Note again that the first option listed is "not...either...or" which is correct grammar even though a few people here are always posting that it isn't correct.
Not/either/or is apparently permitted in English(US) but does anybody know for certain whether it is correct in English(UK)?
I do not know whether our schoolteachers train us not to use it because it is incorrect in UK English or whether we are trained that way because it is merely frowned upon. It's even possible that UK schoolchildren are no longer trained that way, I would be unlikely to know.
When kwiziq says that it is OK in English, I have no idea whether they mean English(US) or English(UK) or whether they have checked that it is valid in both.
"When using ni, you omit the article after ni, unless you're talking about general things and using le, la, l', les."
Agreed, I have seen tisanes on menus in trendy cafes in Canada too.
This never-ending controversy about 'either ... or' after a negative verb where native speakers saying it's correct get downvoted out of sight by those who have learnt their English out of a book, reminds me of a tweet by a British comedian on Twitter at the beginning of January 2022: 'My new year resolution this year is to annoy less grammar pedants'.
Here is the rule: https://www.francaisfacile.com/exercices/exercice-francais-2/exercice-francais-71251.php However, I still do not understand why "je n'ai ni du the ni de la tisane" is wrong when duo says "j'ai du cafe".