book or books
"das kind hat keinen Buch" is translated in : the child hasn't got books. I don't understand why they use the plural here. I would translate it in : the child has no book
That is the more literally translation but it seems kind of wrong when you say it English, everyone would say 'has no books' or just 'doesn't have books'
perhaps this could help, it explains it pretty well : https://forum.wordreference.com/threads/i-have-no-book-vs-i-have-no-books.3046356/
summary of link : " If the question relates to one particular thing, then you usually negate it using no + singular noun. If the question is more general, the plural is more appropriate "
According to my german language feeling, this works for german, too.
"Sie hat kein Buch" . I would think she hasn't one with her.
"Sie hat keine Bücher". I would think she hasn't any at all (eg at home).
I also tried this rule for the examples given in the link, and I always used the same singular/plural in german as was given in the english sentence. except for the last one, where we would use the verb 'arbeiten' instead of a noun.
If I understood you correctly, the english rule is the same. So the two sentences don't match.
There are more than one translations for every phrase, a lot tend to be more idiomatic than direct translations;thats just how language and translation works, in many cases the direct translation is often incorrect or ill-fitting in English, take for example 'me gusta' if we used the literal translation in this course then we would have to type in 'it pleases me' which would never be said in Enlgish therefore it would be illogical and confunsing to teach only direct translations so idoimtic ones are used. I believe that is what happened here, both sentences (buch and bucher) can mean she doesnt have any books. This doesn't make sense in English but German is not English so you just kind of have to let go of the assumptions you learnt from english and open your mind to grammar and stuff that would make no sense if translated into English,
What was the exercise asking you to do? If it's a translation exercise and you give a 'wrong' answer, Duo will give a 'correct' answer that approximates your answer as close as possible. In other words Duo will try to give you the answer it thinks you were going for. Some of Duo's attempts are rather contorted.
BTW: I would translate it 'The child doesn't have a book.' I've never seen Duo complain about using 'do' in negative statements like this.