In the closing credits Anastasia's sisters Tatjana, Maria and Olga is credited with the last name "Romanov". However in Russia female last names have female endings. So it should have been "RomanovA". "Romanov" would be the last name for the male members of the family (as shown for the brother Alexej).
The first noodle factory in the Russian Empre was opened in 1797 in Odessa. So it is very unlikely that Anastasia, being born in 1901, would not have known what noodles are, as she does in the movie. Furthermore being an aristocrat it is very unlikely that she would eat the spaghetti with her hands as it would be against the royal etiquette.
When Megan meets Anastasia as an old woman in 1989 she says that she is "every bit of 83 years old". Anastasia was born in June 1901, which would make her 88 years old in late 1989.
When Rasputin checks his magical map it shows the map of St. Petersburg. In 1917 the city's name was Petrograd.
When Rasputin appears at Disneyland some people in the background don't movie, while others lightly move. It indicates that Rasputin is actually in front of a green-screen with a photo being attached with small CGI effects.
Rasputin was killed in 1916. In the movie he is still alive in 1917.
The beginning of the movie, set in 1917, shows Russia full of snow, which would set the story in the winter of 1917. Lenin is shown being in Russia. However at that time he was in Zurich, Switzerland and didn't arrive in Russia until April 1917.
Czar Nicholas II was killed in 1918, not in 1917 as shown in the movie. Also Lenin didn't kill him personally.
In some of the scenes in Russia the Basilius Church is visible in the background. The Romanovs were in St. Petersburg aka Petrograd in 1917, while the Basilius Church is in Moscow.
Megan is teaching Anastasia how to swing. However earlier she was sitting on a swing with Rasputin.