So that all neatly brings us back to the question of…
How is any of this possible?!
Early theories of an alternative universe version of the Doctor, a possible return of the Meta-Crisis Doctor, or simply being a guest appearance of the Tenth Doctor were all proven incorrect by that game-changing regeneration. Admittedly, this is Doctor Who, so they could still somehow be true, in a way...
The idea of this being a version of Donna from before her memory wipe has been floated. That would explain how she could safely meet the Doctor; as it also would if this was a Donna from a parallel universe. Neither of these theories neatly explains what’s happened to the Doctor; but they’d plausibly explain what’s going on with Donna. Let’s file them under ‘possibles’ and return to the Doctor.

‘Who’ is the Fourteenth Doctor?
Could this strange regeneration be something to do with the Meta-Crisis incident forming a bond between the Tenth Doctor and Donna? While this covers why they’ve been drawn back together, it seems strange that the Doctor has regenerated multiple times since they were last together, without reverting to the Tenth Doctor’s appearance and persona. Neat idea, but probably not.
Another possibility that has been raised is that the Master’s meddling in The Power of the Doctor has affected the Doctor’s regenerations. We’ve just seen the Doctor confronted with previous versions of themselves, perhaps they’ve been drawn back to another former incarnation. That might even solve how the Doctor’s clothes were changed as well. One to file under ‘Distinctly Possible.’
Who knows…
At this point, all we can do is speculate. For definitive answers, we’ll have to be patient and wait.
For a year…
Someone bring us a TARDIS!
Doctor Who will return with the 60th Anniversary Specials in 2023.