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I'm a retired librarian and couldn't resist the question, so I searched some authors for you. Edward Albee has identical twins in "The Ametican Dream", and Tom Stoppard's "Hopgood" is stated in a review that "Twins abound, both real and imagined..."
Couldn't find any twins in plays by Christopher Durang.
The scene you described I couldn't verify.
Thank you for looking into this.
The American Dream by Edward Albee doesn't have any twins. I can't find anything about "The Ametican Dream," I assumed that was a typo. EDIT: Looking a little more there is discussion of a twin, but the twin does not appear in the play.
The plot summaries I've found of Hapgood (again assuming a typo) don't sound anything like the play I'm thinking of.
Edited by randomsurfer

A play by Tom Stoppard, Christopher Durang, or some other absurdist/surrealist writer. Two minor characters are a pair young men, one white and one black, who always appear together and insist to the Only Sane Man protagonist that they're identical twins. In the end it turns out that they are, just not of each other.
Edited by randomsurfer