That would have been kinda cool. Especially if you really hard to work for finding out who it is. Anyone remember the Blade Runner video game? You actually had to do the Voight-Kampff-test in the game, and do it correctly to find out who's a skinjob and who's not.
But of course in Bethesdian Realpolitik it would just be "follow the quest marker to Sturges' locker and read his diary he's totally a synth lol" (granted, the Far Harbor Vault quest was actually slightly more than that, and the end was even surprising) instead of an involved hunt with several suspects and only your own judgement on the end. Imagine you'd retire on of the suspects just to find out that you were wrong, with all the repercussions from the faction. But that would have been "consequences" and stupid shit like that.
Which reminds me of what I've said before, the game should decide randomly at start if the player character is a synth or not, and the only way to find out would be to kill your doppelgänger at the end of the game (with your doppelgänger "learning" from your actions during the game and reacting accordingly. Like, be a total douchebag in your playthrough and the doppelgänger will be a pillock as well, be a smartass all the time and your doppelgänger will be supremely sarcastic during your conversation).
Heh, how about you can let your doppelgänger kill you as well, and play on after the end as your doppelgänger
So meta.