The Statue of Liberty is shown as being already green when it should be brown. The bronze didn't deteriorate to green until around 1900.
Félicie is wearing 1950s denim shorts in some scenes.
"The Nutcracker" was first performed in Saint Petersburg in 1892, 3 years after the Eiffel Tower was completed.
Odette nicknames Félicie "Sherlock Holmes". The first Sherlock Holmes story was published in 1887 (possibly the year the movie is set) in an English magazine and wouldn't have been known in France until sometime after.
The actual Quebec flag is also put up at the Breton bar scene. But it was only created as such in 1948.
A nearly completed Statue of Liberty and a half built Eiffel Tower are shown to be in Paris at the same time. In reality, the Statue of Liberty opened in New York in 1886, and the Eiffel Tower started construction only in 1887.
The performance of The Nutcracker has a scene which is found nowhere in the play. In the real play, Clara spends the majority of her scenes in a nightgown (rather than a pancake tutu), and never duets with the principal ballerina (the Sugarplum Fairy).
The movie seems to be set sometime between 1885 and 1889. While a motorcycle powered by an internal combustion engine was invented in 1885, Mr. Luteau's machine seems to be a design from 1895 at the very earliest.
While at the Breton bar in Paris, the 'Gwenn ha du' flag is shown. (It vaguely resembles a B&W version of the USA flag.) But it was only created in 1923 (and became popular from the 1970s on). If a Breton flag had to be put up at the end of the 19th century, it'd have been the 'Kroaz du' or the plain ermine flag.