Hi,
Whilst the Apple ID you use does not have to be the same one you use in the iCloud Settings it does have to be a valid iCloud ID.
Put another way - All iCloud IDs are Apple ID but not all Apple IDs are iCloud ones.
Therefore your could use an older @mac.com name that has not been linked to iCloud and it will not Register with the iMessages servers.
It could also be that the ID is use is not working in iCloud either.
You have given no indication of whether this Apple ID works in iTunes, iCloud, iBooks or in the Photos/iPhotos App at all.
You also have not indicated whether you have a Mac (You have posted in the iChat area which is an Mac only app that was replaced by Messages on the Mac at Mountain Lion (OS X 10.8.x). )
If you Log In here https://appleid.apple.com/ does it indicate that there is an issue with your Account ?
As it is on two devices and each login from each device is unique based on the Serial Number as well as the Name/ID and Password combo the issue is with the whole Apple ID rather than the apps on each device.
7:58 pm Friday; January 8, 2016
iMac 2.5Ghz i5 2011 (Mavericks 10.9)
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