Front end Airbnb customer service has only the foggiest notion of what taxes they remit. I wouldn’t put much stock in what they tell you. The two times I’ve had tax questions the front end rep has passed it on to the Tax & Legal team (I think that’s what it was called) and then I never hear back from them again. (At least 6 months waiting since my last question) So good luck getting an answer!
You’re probably better off checking the math: You can check your transaction history, then hit the “gross earnings” tab. This will have a column on the right showing what tax amount they collected/remitted. See if that amount aligns with the State tax only or if it was enough to cover State & city taxes.