Redundancy
The original question, for A-320, states that the Autoland status is Cat3 Single for a single engine approach, then questions whether that has to do with one of the electrical sources being the APU generator instead of the generator of the broken engine.
No, the status is Cat3 Single because you have only 1 engine operating!
Cat3 Dual means the system is fail operational: no single failure (from that start situation) will preclude the system from completing the landing automatically.
Note: if that single failure occurs below Alert Height. If the failure happens above Alert height, you will see a reversion of Autoland status to Cat3 Single.
If you start the approach with already one engine out, then how can you expect the system to cope with a failure of the remaining engine? Thus when flying single engine, the system is not fail operational, but fail passive.
The misleading concept may be the point that in this Autoland status discussion, the failure of the second engine would be a “single failure”, because you have to look at the situation with which you started the approach, namely with one engine already out – from that initial situation, a “dual engine failure situation” is only one step away, so only “a single failure”.
Now, for a 2 engines operating situation, but with 1 engine driven generator u/s and replaced by the APU generator – indeed, there you will also see Cat3 Single, according to FCOM. The reason for that, not sure,like MCDHU stated earlier, there may be some common relay in the system.