It has been awhile since I read a couple of biographies of Stalin and his Terror. Stalin's history during the time of the play is rather fuzzy. He didn't like facts of his early life to be known and suppressed them. Until he came to power, a lot of people didn't take much notice of him at all. One communist of the period described him as "just a grey blob". In many photos he's simply the guy standing just behind Lenin. Once Lenin started having his strokes, Stalin took power quite quickly, leaving people scratching their heads and wondering "who is this guy, and why haven't I noticed him before (and why am I being arrested?)?"
Also, I don't think he had adopted the name of Stalin until around the time of the revolution. I think he was still using his real name of Josef Djhugashvili. He later started calling himself Koba (after a Russian folk-tale character), then Koba-Stalin, and finally Josef Stalin.
This play sounds like a 1970's play by Tom Stoppard called Travesties, which featured an Englishman remeniscing about living in Zurich at the same time as James Joyce, Lenin and Dadaist Tristan Tzara.