Colin Baker played a Gallifreyan guard named Maxil in a Fifth Doctor story before being cast as the Sixth Doctor, and that was never mentioned again, despite how much fun a story dealing with that could be. Just think about it: two incarnations of the Doctor, working at the same time in the same place, but working to fight off threats just different enough that neither can reveal their identity to the other, but every once in a while, it could overlap their stories to show what the other was doing in the previous episode/other side of the conflict.
Of course, since the Doctor Who-canon characters played previously by Peter Capaldi were indisputably human, and one of them died, that's not something that can happen now, either, but there's always a possibility of an actor allusion at some point.

From the few examples we have, all of the Clara copies shared many of her personality traits. Amy is very little like Clara, even if it can be assumed that the process that threw Claras across time-space made some of them look like different people.

At any rate, what would distinguish Amy in that way? Because she was the Doctor's companion? Would your idea being true mean that every companion was also Clara, or would every woman who helped the Doctor secretly be Clara? Where would it end? Would every human in the universe secretly be Clara Oswald? Or would every single humanoid, regardless of species or gender, simply be versions of Clara that strayed farther and farther from the template and had their physical and genetic structure warped as they strayed from Clara Prime?

But if everyone is Clara... no one would be.

Clara is... well, Clara. In all of her appearances in the Doctor's timeline, she always looked like Clara Oswin Oswald. She isn't a face-changer, multiple people can attest to that, so how does having Karen Gillan play a soothsayer in Pompeii make Clara be there?