Translations are not word-for-word.
Because Duolingo is essentially school, you translate it into standard English: I don't have anything in my bag.
If the context were more informal and you were translating into a dialect that has negative concord, then it would be appropriate to say "I don't have nothing in my bag" or even "I ain't got nothing in my bag."
On Duolingo, though, stick with standard/academic/business English.
Once again, people, language is not math. They don't "cancel each other out". Many languages require negative concord, which can be thought of as similar to how adjectives agree with nouns. English is somewhat unique in its stigmatization of negative concord as the so-called "double negative".
Given that the Italian has "Non ho" at the beginning of the sentence, Duolingo is likely expecting you to include the translation "I do not have" at the beginning of yours. "I have nothing" means the same thing in English, but it is not an accurate translation of this Italian.
No. Double (or even triple) negatives are normal and common in Italian (and other Romance languages, for that matter). "Ho niente..." might work colloquially, but the normal way of saying it is "Non ho niente...". See here: http://italian.about.com/library/fare/blfare141a.htm
It's called "negative concord". Not all languages consider the "double negative" to be a bad thing.
It's called negative concord and a lot of languages have it, including English. It's just that not too long ago some English-speaking people decided that language "ought" to behave like math and logic (spoiler alert: it doesn't) and decreed that negative concord is "wrong" (language does not work by decree).
Think of it as similar to how adjectives need to agree with nouns.
It can also translate as Anything. I do not have anything In my bag. Italian does use double negatives often but in this case it is saying that it does not have anything. niente [ˈnjɛnte]
PRONOUN (nessuna cosa) nothing (qualcosa) anything non... niente nothing ⧫ (espressione negativa) + anything https://www.collinsdictionary.com/us/amp/italian-english/niente
nothing Synonym: nulla Cosa fai? Niente. ― What are you doing? Nothing. anything Synonym: nulla Hai bisogno di niente? ― Do you need anything? Non voglio niente da mangiare. ― I don't want anything to eat
Translation is not about taking each word individually. And it's called negative concord, which is a thing many languages have, even many dialects of English (although it has disappeared in standard English). Most people who call it a "double negative" are just irrationally stigmatizing language for not being like math.
Duolingo is an American company and the volunteers who put this module together are American. It stands to reason that they are more familiar with how we speak here in the USA than with how people in other English-speaking countries talk. If your valid, correctly-spelled translation is marked wrong, you need to flag it and report "My answer should be accepted" before you move on. If enough people flag the same answer, the team will consider adding it to the database for this prompt.