It's on the English side, no double negatives. What exactly was the sentence that was not accepted? did you report?

"I have neither wine nor beer." should be accepted now

I haven't tried it as an answer, but "either... or" shouldn't be accepted here because that's incorrect English.

Well, I guess it would be if the sentence was simply "Nie mam wina", but here, with 'neither... nor'?

Hmmm... Well, ani... ani does show emphasis, so I guess your version makes sense. Added.

Advice: don't post walls of text in the comments that nobody will read. Simply write what's exactly wrong and leave the links to the sources.