The explanation
There was a glitch (May 2019) with OverDrive accounts that OverDrive is attempting to fix by requiring every user to re-authenticate their library card information in each of the apps/websites used when attempting to place a hold or borrow a title. You should only have to do it once in each situation, but if you haven't used OverDrive in some time, you may not have experienced this yet.
The solution
On the OverDrive website
Log in with your card number and PIN, and you should be good to go. Please note, if you're going to download and read a title in either the OverDrive or Libby app, you'll need to verify your card number in those apps in order to view the checkout.
In the OverDrive App
Go into the app settings for Account and sign out if you are not already, then log in with your library card number and PIN.
Please note: the OverDrive app will be discontinued on May 1, 2023. As Amazon has not yet accepted the Libby app to the Amazon Appstore, the OverDrive app will remain available for Kindle Fire users to download and access until further notice. Users with Kindle Fire devices can also sideload Libby manually following these steps.
In the Libby app
When you borrow or place a hold, you'll be prompted to verify your card by entering your PIN (and possibly your card number).
If you now find that you have two cards listed in Libby, here's how to remove the extra card.
Still having issues? You can Get Help Using FVRL's Online Resources, or try the OverDrive Help site or the Libby Help site.