Thanks Tebogo, but I chatted with someone and got sorted.
When the Apple TV lost it's connection, there was no need for me to remove it under Manage Devices to re-link it. I only needed to go through the linking procedure again, and it was recognised as the same device.
Our Apple TV will be registered under our DSTV Now devices and working normally. Then one day, we'll turn on the Apple TV and it will ask us to link the account account (Enter TV Code to link to account).
We are then forced to remove the device under manage devices and re-link it.
It just happened again tonight, but since this has happened twice this year already, we have no more removals available to re-link it.