It seems that something else is going on here that isn't related to the serial number. My serial number presents itself when I click on "About This Mac" yet I am still having this issue. I have tried all of the solutions presented by Colourist, and there has been no improvement of my situation.
At the moment, I am unable to sign in to any apple application (messages, App Store, iTunes, iCloud, Etc.) on my Mac. I can sign into each application just fine on my iPad. I do not get any error, I just get a "loading" wheel forever which means I have to force close the app in order to close it. If I deliberately put an incorrect password in, I get an error right away stating that the password is incorrect. Therefore, the app is talking to Apple, and my computer is running a check. For some reason, my request for sign-in is not being authenticated, even though my date and time settings are correct, i've deleted all caches mentioned above, I have a serial number showing, and I have properly authenticated on this computer prior to El Capitan.
I have seen that multiple people have had this issue, but it doesn't appear to be widespread enough to make front page when googling this issue. There have been no actual solutions presented.
Does anyone else have any other ideas? It seems like it's something wrong on Apples end... I really wish they would take a look at this.....
I don't have AppleCare, but I am contemplating calling them anyway because this doesn't appear to be an issue with anything I have done to this computer. Perhaps if enough of us call AppleCare requesting a solution they'll get the idea that something is wrong.
But for now, no fix in sight, and no idea what the actual issue is...