Since the Statue of Liberty is shown to be under construction in Paris, we know that the story takes place in 1886 at the latest. Yet at one point, Odette nicknames Félicie "Sherlock Holmes". The first Sherlock Holmes story was published in 1887;
The movie shows the construction, in Paris, of the Statue of Liberty, at the same time as the construction of the Eiffel Tower. The Statue of Liberty was dedicated in New York in 1886, while the Eiffel Tower, in its current location, did not commence construction until 1887.
In the movie, it shows both (in Paris) a nearly completed Statue of Liberty and the start of the construction of the Eiffel Tower. In reality, these two world'famous landmarks were never in the same continent at the same time. The Statue of Liberty was given to the United States in 1886, before the start of the Eiffel Tower's construction, which commenced in 1887 (and was completed in 1889). Both were constructed by Alexandre-Gustave Eiffel. But Eiffel only started construction work on the tower bearing his name after the Statue of Liberty has been shipped to the United States.
While at the Breton bar in Paris, the 'Gwenn ha du' flag is shown. 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.
The time line of this movie suggests it is somewhere between 1885 and 1887. The motorcycle powered by an internal combustion engine was only invented in 1885. Yet the motorcycle shown in the movie as ridden by Mr. Luteau looks more like a motorcycle design of the late 1890s or early 1900s.