Why is "forbidden" wrong?
Yes, I think that is actually better in this case.
Why is "I am being prohibited from drinking wine" wrong when the hint given for Bị is 'being'
"Bị" is Vietnamese word used to mark passive voice. It is the equivalence to English "to be" in this case.
as Huy has said, bị indicates the passive voice. the difference between their translation and yours is mostly the progressive tense used in your answer.
forbidden is better than prohibited