With these types of sentences, the parts following "neither" and "nor" should be equal. If one contains a verb or not, the second one must be the same.
With your question, the correct way to say it is "I neither drink tea nor drink milk". Both neither and nor has a verb. But this is not the Russian phrase ("Я не пью ни чай, ни молоко.") is asking for, since it talks about two things that you don't drink and the correct translation is "I drink neither tea nor milk".
In your response, the words "neither" and "drink" need to swap places. Your sentence turns the word milk into a verb, essentially meaning: "I don't drink tea, and I also don't milk," - which doesn't really make sense in this context. (Perhaps on a farm, in the context of a discussion about which farm responsibilities you handle? "I just clean the stalls, I don't milk [the cows].")
Actually, the direct object uses accusative. The accusative for most masculine and neuter nouns looks exactly like the nominative. If you use a feminine noun (like вода), you will see the difference: Я не пью ни чай, ни молоко, ни воду. (Note that you can make a list of any length using ни ..., ни ..., ни ...)
As a native Russian speaker, I would like to know this too... gramota.ru explicitly says  that ни... ни requires genitive case. However, my intuition tells me that this particular sentence should use accusative. Who is right?
Well... this is not the negation where you should use genitive. Grammar resources for native speakers can confuse you because they always rely on some "native" background.
If you said "I have neither tea nor milk", that would be "У меня нет ни чая, ни молока" - with the genitive case.
As a second thought... yes, you can also say "Я не пью ни чая, ни молока" (using genitive), but this will be less common.
I'm starting to think that genitive is "more correct", but accusative is used in informal speech (and is now acceptable in formal too). See http://www.evartist.narod.ru/text1/66.htm, §201.1.2) он не пьет вина и не ест мяса (Gorky) and §201.3 "Факультативное употребление родительного и винительного падежа при переходном глаголе с отрицанием связано со стилистическим различием: конструкции с родительным падежом характерны для книжной речи, конструкции с винительным падежом – для речи разговорной. Ср.: Я не умаляю чьих-либо заслуг... (Горький). – Так и умрешь, не выговорив это слово (он же)."
The first time through this review lesson, I got the English sentence, "I drink neither tea nor milk." I answered, "Я не пью ни чая, ни молока," and the response was "Awesome!" I keep a written list of the questions and answers for future reference, so I made a note of it.
About two minutes later I got the reverse as a listening exercise. I was confused because it sounded like she was saying "... ни чай, ни молоко." But I had just entered the genitive earlier, and it was awesome, so I entered it the same as above. WRONG!
Very inconsistent! And I thought that negation always took genitive, so now I am just confused. As usual, zero flexibility.
There is much research on the subject. The papers show only statistical trends for preferring one form over the other in specific circumstances. Whereas using the Genitive for the negative object of the verb that would take Accusative in a positive statement was widespread in the 19th century, it is not the case nowadays for most of them.
Я не пью ни чая, ни молока. is acceptable but a bit old-fashioned.
Rozenthal (who died 20 years ago) was more moderate in his phrasing in the page you linked to, saying that the presense of "ни" is usually associated with genitive (but then he provides the example where this trend is broken because of using specific objects like people's names)
I am on the duo app, does duo learn us about Russian cases and how to use them or do i have to look else where, i practice everyday and everyday i get things wrong because endings have changed or different cases, it says we learn by mistakes but for me this is negative ,i spend more time looking in the comments looking for answers than i spend learning and practicing
I think you might be confusing чья and чая.
The latter (чая) is the genitive form of tea (чай).
On the other hand, чей (masc.), чья (fem.), чьё (neut.) and чьи (plu.) are all forms of the question word "whose."
There is no case form of tea that is чей. Is it possible maybe that you are remembering the sentence, "Чей чай она пьёт?" meaning "Whose tea is she drinking?" In that sentence, whose is "чей" because tea (чай) is masculine in Russian. Hope this helps!
I translated it as "I drink neither tea nor milk." It accepted the answer, but states that the correct one is "I don't drink either tea or milk." which as far as my English knowledge means that the character in the sentence drinks one of the two - so it's a wrong translation of the Russian sentence.