There are some articles about tax collection, i mentioned them below.
The article about "tax collection and remittance by Airbnb" contains a link to the state of Alabama, which explains also taxes collected for specifically Birmingham.
Unfortunately it does not mention to whom Airbnb exactly sends the taxes, which seems to be your issue..
I fear only Airbnb support can provide that answer to you.
The first article explains how taxes are collected and where you can find "tax collection eligibility" (if any) on your listing(s)
How tax collection and remittance by Airbnb works
https://www.airbnb.com/help/article/1036
Areas where tax collection and remittance by Airbnb is available
https://www.airbnb.com/help/article/2509
@Lance0