When Does the Animated Film "Anastasia" (1997) Take Place?
"Anastasia" is a historical fiction fantasy animated film-musical written by Eric Tuchman with Susan Gauthier, Bruce Graham, Bob Tzudiker, Noni White, and Carrie Fisher [adapting "Anastasia" (1952) by Marcelle Maurette] that released November 21st, 1997. So when is "Anastasia" set?
It takes place in the year:
We know this because when Vladimir Valya "Vlad" Vonitsky Vasilovich is seen forging passports for the main characters a third into the movie and at least one of them reads, "LE PASSEPORT EST D ÉLIVRÉ LE March 5 1921" and "LE PRESENT PASSEPORT EST VALABLE LE March 5 1929" (French that translates to "THIS PASSPORT WAS ISSUED ON THE 5th of March 1921" and "THIS PASSPORT IS VALID ON THE 5th of March 1929"). So it must be between 1921 AD and 1929 AD. Also, while in Paris about three-fourths through the story the characters buy clothes at the Chanel store, which was founded in 1909 AD and began selling clothing during World War I, i.e. between 1914 AD and 1918 AD. So it must be after the end of the war in November of 1918 AD.

1927 AD
We know this because when Vladimir Valya "Vlad" Vonitsky Vasilovich is seen forging passports for the main characters a third into the movie and at least one of them reads, "LE PASSEPORT EST D ÉLIVRÉ LE March 5 1921" and "LE PRESENT PASSEPORT EST VALABLE LE March 5 1929" (French that translates to "THIS PASSPORT WAS ISSUED ON THE 5th of March 1921" and "THIS PASSPORT IS VALID ON THE 5th of March 1929"). So it must be between 1921 AD and 1929 AD. Also, while in Paris about three-fourths through the story the characters buy clothes at the Chanel store, which was founded in 1909 AD and began selling clothing during World War I, i.e. between 1914 AD and 1918 AD. So it must be after the end of the war in November of 1918 AD.
More specifically, comrade Phlegmenkoff says to Anya (loosely based on the real-world Grand Duchess Anastasia Nikolaevna), "For the last 10 years, I've fed you, I clothed you"..., referring to how long Anastasia has been in her care in the "PEOPLE'S ORPHANAGE", which must be roughly the time since the end of the narrative's prologue.
1917 + 10 = 1927.
Furthermore, the 1916 AD (or 1913 AD)-1917 AD prologue ends with a superimposed title card reading "ten years later".
Additionally, there's the prologue in which giant circles of lightbulbs spelling out "1916" can be seen under the much smaller number "300". Furthermore, this prologue almost immediately includes narration from the real-world Dowager Empress Marie Feodorovna clearly stating: "The year was 1916 and my son Nicholas was the czar of Imperial Russia!" The chandelier in the palace has been altered with the number "300" and Dowager Empress Feodorovna's narration continues on to say, "We were celebrating the 300th anniversary of our family rule." In the real world, the Romanov Tercentenary happened in February of 1913 AD to celebrate the real-world election of Russian Tsar Mikhail I at the Zemsky Sobor in January 1613 AD. (Electric lightbulbs did spell out "1613–1913" at the real celebrations.)
Either way, fictional events follow to precipitate the Russian Revolution and the evacuation of the Romanov royal family is dramatically depicted in the end of the prologue. In the real world, this evacuation was orchestrated by the temporary government and occurred in August of 1917 AD; then the family was executed the following year.