The bulk of the story is set in an historical past - real history, as accurate as I can make it. The twist is that the historical MC has a mental illness (there are genuine clues in history), and the other part of the story is set in the present day, where his doppelganger has the same mental illness. It's a bit like Anya Seton's Green Darkness.
Not enough information to decide a genre. Maybe you're trying to decide fiction from non fiction? Which I don't get. Many stories that are fiction are based or inspired by actual - historic events.
Yeah, this could be any genre or none at all. I wouldn't get hung up on that though, Shadowfax. If your story is to fit any genre, it will come out as the story is told. Or maybe not at all. Take Mitchell's Cloud Atlas or Morgenstern's Night Circus. I haven't once seen either of these in the SF or fantasy sections of bookshops. They've been marketed as 'literary' and done very well for being so ... so I wouldn't worry too much about it.
If it’s a very straight story about mental illness, it’d most likely fall into the drama genre. However, the mention of the doppelganger has me assuming that these two people/stories somehow overlap. Perhaps, somehow, these two characters may even meet. If that’s the case, you’d most likely be talking about sci-fi. But then, as mentioned, any manner of things could happen in your story. These events could make the story a horror. Or a romance. Or...