I do a lot of travelling Peloton fittings, and yes, saddle position from road to Peloton stays the same. Handlebar position can be awkwardly long for shorter riders, but I don't move the seat to compensate.

Is Peleton saddle proprietary or can it be switched to a "normal" one?

I've seen a lot of Peloton peeps who also ride a road bike somehow manage to change out the Peloton saddle for a bike saddle they like. I am tempted to do it. After I was just riding my bike outside (finally, after besting a lot of nerves) and also on the trainer, I realized the Peloton saddle is terrible for me. Sigh.