Other mistake: In the song "A Rumor in St. Petersburg," according to the closed captioning, the characters sing the words "no proof of fact." According to the lyrics in the soundtrack booklet, they are actually singing "or through a crack." And if you listen, it's actually what they are saying.
Factual error: Anya receives a sign from the dog to go to St. Petersburg when she leaves the Orphanage in the mid 1920's. St. Petersburg was renamed to Petrograd in 1914 at the start of WWI and again to Leningrad in 1924 following the death of Lenin. The sign should have said "Leningrad". Aside from signs, the whole movie mentions "St. Petersburg" multiple times. There's even a song about St. Petersburg. Even if every old sign could not be changed, there is no reason the people would not use the city's new name. The Bolcheviks would not have looked at them kindly if they didn't.