In real life, Olga really did say that Anastasia's drawing looked like a pig riding a donkey. This was stated by Anastasia in a letter to her father, and the image used in the movie is an actual reproduction of the original picture.
The portrait in the ballroom of the whole family includes a spaniel. The spaniel existed. The spaniel, named Joy, belonged to Anastasia's brother, Alexei, and was found alive at the house where the family was killed. Anastasia's own dog, Jimmy, did not survive.
When Meg Ryan was offered the role of Anya, she could not decide if she wanted to accept it or not. Upon hearing of Ryan's indecision, Fox took an audio clip of Ryan talking in Sleepless in Seattle (1993) and created a short animated sequence of Anya speaking the lines. They sent the clip to Ryan, and she was so impressed that she changed her mind and accepted the role.
In real life, Gregori Efimovich a.k.a. Rasputin was a very controversial figure who, in fact, was the Romanov's advisor and Tsarina Alexandra's most trusted confidant. Rumor has it that Rasputin told the Tsarina he was about to be assassinated and that if one of her relatives killed him, all the Romanov family would die within a year. While of course these facts were too dark to be included in the movie, there is a reference: during the song "A Rumor in St. Petersburg", an old woman tells Dimitri to buy "Count Yussupov's pajamas", while offering a pair of ragged clothes. Yussupov, who actually was a prince, really existed, was indeed related to Alexandra Romanov and was the one who killed the real Rasputin, along with a group of noblemen.
When Anya returns to the palace in St. Petersburg and is in the ballroom you can see the painting of the coronation of Alexandra and Nicholas on the left hand side being the first picture, which is a real painting.
The drawing the Empress holds when she and Anya are reminiscing (the same one we see little Anastasia give her at the beginning of the movie) is a picture the real Anastasia had drawn for her father in 1914.
The character of Dimitri was based on a European prince who vouched for Anna Anderson's identity as Anastasia. The prince had only met Anastasia once and during her childhood, though, so he was not considered a very credible source.
The Parisian bridge on which the confrontation between Rasputin, Dimitri, and Anastasia occurs is the Alexander III bridge, named after the real Anastasia Romanov's grandfather on the occasion of his state visit to France in the 1870s.
The character of Vladimir is based on Count Vladimir Frederiks, Tsar Nicholas' Chief Court Minister. He was very close to Tsar Nicholas and his children and remained in Russia for years after the revolution, wearing his court uniform in protest.
Bernadette Peters was not pleased with the design for her character. According to Gary Goldman, Peters was very physically fit at the time of production and was disappointed that Sophie was drawn so heavyset.
Liz Callaway was called at the last minute by Flaherty and Ahrens to substitute for a singer who couldn't make the recording session of the temp tracks for Fox. Her tracks of the songs were liked so much they led to her subsequent casting as the singing voice of Anastasia.
As is the case with many 20th Century Fox Films, the film cans for the advance screening prints and show prints had a code name. Anastasia was "The Train". There is a climactic train wreck in the film.
[the music box]
Grand Empress Marie gave the music box to young Anastasia so that she'll have something to remember her by before she returns home to Paris. Only the two knew the music from the music box as it is the "Once Upon a December" lullaby that Marie sings to Anya at bedtime. That music box, as well as Anya's memorization of its music, were to proof to Marie that Anya is her long-lost granddaughter, Anastasia.