I had the dreaded auth loop (but in Big Sur)... but was able to work around it by changing my Web Browser to Chrome (via System Preferences, General)... then was able to de-auth audible from Books... and got a message that said successful from Books (although it still indicated, under the Auth menu, the deauthorize option and not the authorize option... so still kinda broken). Anyhow, then quit Books (completely, not just the window but 'Quit' from the menu), restarted, then tried to play a book that was having this problem... usual prompt came up, followed it (into Chrome, not Firefox), logged in and clicked the Authenticate button and, boom, this time it worked.
Aside: It should be noted that what wasn't playing in my library was older audible audiobooks that I downloaded back before Amazon took over Audible... I still had my original audible account (which had a userid, not an email address, and a password)... and the amazon login allows you to select login with 'username' which is this old audible.com account name. So I had to use *that* account to authorize the set of books that were not playing (the other books were purchased through Books, aka: Apple, and those were fine). Anyhow, you may need to insure that you authorize the correct Audible account (if they are older audiobooks).
Aside: it is annoying that I have, effectively, 2 audible accounts now with my old Audible account and the new Amazon based account, with books purchased under the old and new essentially "separated"... and then there's the books purchased under Books via the Apple link. It's a bit maddening. Perhaps Amazon, at least, can get those old Audible books migrated and auth'd under the newer Amazon based account via my Amazon login/etc. I won't hold my breathe.
This may not do much for anyone, but tossing it out here in case it helps... as this is frustrating.