While it is a historical fact, that Sisi didn't raise her daughter Sophie first, it was not done by Archduchess Sophie but by Baroness Aja Charlotte Welden. Unlike shown in the movie she didn't get to raise it.
It is announced tat Sissi's daughter was baptized on the name "Sophie Anastasia Amalie Elisabeth Franziska Stephanie Caroline Maria". However her real name was Sophie Friederike Dorothea Maria Josepha.
Sissi told Franz Joseph that the people of Vienna were very happy to see her that they almost ripped her dress. In real life Empress Elisabeth wasn't that popular in Austria. She was more loved in Hungary and less in Austria.
During the ball scene with the Hungarians the "Emperor Waltz" by Johann Strauß Jr. is played. This waltz was released in 1889. The scene however takes place in 1854/5.
The coronation of Empress Elisabeth to be the queen of Hungary was in 1867. Only one scene before that Sissi's daughter was brought back to her (which never happened in real life). However Sisi's daughter Sophie died in 1857, 10 years before the coronation.
When Sissi is traveling to Budapest for the coronation, several herds of pigs are seen. All these pigs belong to the widely known "large white race", that was selected in England in 1868 and subsequently did spread throughout Europe. The coronation took place in 1867, when Hungarian pigs were smaller and hairy.
Sissi and Franz drive for hours to get to Hungary, but the crowning take place in the same church, as the wedding in part 1 (St. Michael's in Wienna). When they step out of the church they are back in Hungary. You can see the pretty specific altar section in both part 1 at the wedding and part 2 at the crowning.