You couldn’t fully write the play around them if you wanted to to put it on reliably, but you could probably take some interesting shortcuts. “Oh, the protagonist is supposed to wake up from a nightmare in a nervous sweat? Okay, cut to black while they’re in the nightmare scene; moments later, open the lights on another part of the stage where the identical twin actor is fully dressed in pajamas and wakes up in bed.”
As long as you juggled this right, you could have a character that teleports and changes costumes instantly. I’ve never done theatre, but this sounds versatile.
Aaron and Shawn Ashmore are twin actors, I know that although Aaron actually played both of a set of twins in My Brother’s Keeper, Shawn is actually in some scenes.
It was more of a special effects shortcut, but it’s twins playing twins, so I don’t know if that counts!