Same problem Catalina 10.15.5. I thought at first it was because the audiobook I decided to stick on a new laptop was airdropped onto it from a different laptop and not downloaded fresh from my Audible account (although that was never a problem back when for instance I went to a new laptop running Mojave from one running El Cap).
Anyway just to be sure that was not the issue, I went to Audible site and and downloaded one of my purchased audiobooks directly to new MBA and stepped through the auth process again. Again the book would not play.
I notice in the Books app under Store menu, under Authorizations, the "Deauthorize Audible account" remains greyed out, so whatever is happening on Audible's end as "activation" is definitely never landing via Catalina in the right place on this laptop for Books to find it.
And in Books under that Store menu, Authorizations, if I say "Authorize Computer" for the heck of it, it just says I've already authorized this computer and reports the number of computers I have authorized, which number is still correct.
Gotta love how Audible's site suggests to Mac owners hey why not go mobile and listen to your Audible books with our mobile app "Audible". And then they provide perfunctory how-to make it happen on a Mac, but it's a dead end at least for me.
Hey guys I just want to use Books or "Music" on my new MBA, m'kay?
Not having fun at that impasse. So I've had it for today and will move on to trying to figure out what to do about my music libraries tomorrow and get back to Books if and after I survive the Catalina experience of "Music".
Have to say the new machine itself is a trip though. I'm so pleased with it. Just wish I could graft a few pieces of Mojave in there at least temporarily.