Hi Eric and DavidNYC,
A couple of things.
In Messages (before you even tackle the Start up Screen) you technically have two accounts already.
The iMessages one and the Bonjour one.
Neither can be deleted and the iMessages one needs an Apple ID
Also neither can be added the the Mail, Contacts and Calendars pane in System preferences.
After you set up the Messages account or via the Mail, Contacts and Calendars items you set up what used to be the iChat type accounts (AIM, Jabber and Yahoo).
If you migrated a Mac User Account it is possible you will have older iChat Accounts already added by the time you can look in Messages > Preferences > Accounts.
@DavidNYC
There are some historic causes:-
FaceTime, Game Center, Messages: Troubleshooting sign in issues
Not mentioned in that article is the issue of the missing Serial Number which I see you mention below - I don't think this has raised it head again.
Linc is probably right in his evaluation that the shortest fix time-wise will be a restore of the OS.
Deleting com.apple.imservice.imessages.plist and com.apple.imessages.bag.plist will remove the ID and the linked "Aliases" the mac version has for iMessages - causing you to have to reset up the iMessages account.
You could try deleting the Auth-tokens in the Keychain Access app for FaceTime and Messages but I am stuggling to see why they would have become corrupt and would be hard pressed to create a proper argument as to why you should do this.
Some people have suggested that you (anyone) should visit the Apple ID site and make sure your account has info such as the iPhone Number.
Personally I have not had to do this. I have two iPhones and use two different Apple IDs and do not allow the Mac Version to add the second iPhone or it's Apple ID as an "Alias".
My two apple IDs are linked to the same iCloud account.
However it does not have to be an iCloud ID.
Having said the last two items it is possible that Apple have done something relating to the "security" of using an Apple ID - similar to the Serial Number - but possibly based on info held at Apple.
However the info on the OS X 10.8.5 suggests Messages is untouched and there is no suggestion of any changes to support apps for Messages
An alternative to Restoring the whole OS is to download and run the COMBO version of the Update over what you have.
8:59 pm Tuesday; September 17, 2013
iMac 2.5Ghz 5i 2011 (Mountain Lion 10.8.4)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
Couple of iPhones and an iPad
"Limit the Logs to the Bits above Binary Images." No, Seriously