Russian --> English translators often have issues of article placement. In English, it can make sense to say "I do not have a cat or dog/a cat or a dog/a cat nor a dog"
Not sure how Duolingo reconciles this within this context, but it is good to get into the habit of translating Russian to English with appropriate contextual article placement.
I have neither a cat nor dog? I neither have a cat nor dog sounds more accurate, yet that's not accepted