AirPort Express Unrecognized Apple ID

The first question here would be whether or not you have "upgraded" your Mac to use the Mojave operating system. If you have, you might not be aware that Apple has dropped the Back to My Mac feature from Mojave, so that may be the reason why you are seeing the Apple ID error message.

So, you have moved up to Mojave, you will need to disable the Back to My Mac function to clear the Apple ID errors.

Even if you have not moved up to Mojave, you will still need a Mac that is running a current or recent operating system to be able to access the Base Station settings for the AirPort Express......to reenter your Apple ID information and password.

The reason......AirPort Utility on an iPhone/iPad does not have the ability to access, change or edit your Apple ID settings on an AirPort router, so you will need a Mac to check and possibly adjust the Base Station settings in AirPort Utility.

Please post back to clarify if you have a Mac running Mojave or another operating system and we will continue.

Would a reset be a viable solution?

If Back to My Mac is not enabled on your AirPort Express devices, I doubt that resetting them would help at all, so that would be a last resort with an Apple ID issue. Only Back to My Mac uses the Apple ID info on an AirPort router, and if BtMM is not enabled, there are no other Apple ID settings to check on the AirPort or in AirPort Utility.

If this is an Apple ID issue, I would work with the Apple ID settings on your Mac.

Open up System Preferences > iCloud and look around there first. If the box next to Back to My Mac has a check mark, uncheck that box. Then Sign Out of iCloud, then Sign In again and restart your Mac. Check to see if that helps.

If the error still appears, then you need to check your Apple ID Sign In / Sign Off in the App Store on your Mac. Signing Off, then Signing back In might help. Restart the Mac as well.

If all else fails, then try resetting the AirPorts back to factory default settings and then set them up again. Use a different password when you set them up again, in case the old password storage has become corrupted.

Would a reset be a viable solution?

Yes, performing a "factory default" reset on both base station will clear out any saved passwords and return the base stations back to their "out-of-the-box" configuration.