The film ends with the real Ivan IV being sent to his proper time, while Ivan Bunsha escapes back to the modern times when the guards figure out he's an impostor. Now how exactly is the tsar supposed to prove he's the real deal before he's disposed of à la Vlad the Impaler. Granted, the whole thing is All Just a Dream, but still. Of course, given who Ivan the Terrible really was in Real Life, he may have deserved it.
Given how quickly (over the course of several hours) did they figure it out, there's probably someone who can tell whether the tsar is real or fake. Also, though identical, they had very obviously different speech patterns, mannerisms and personalities, so, probably, it just took the tsar to roar at the guards to be recoginzed as true Ivan Vasilievich.