Around the World in Eighty Days takes place in 1872; Bass Reeves did not become a US Marshall until 1875 and served in Western Arkansas until 1893.
Bass Reeves is portrayed as being in his fifties, with a graying beard. In reality, Reeves was 34 years old in 1872.
Marshal Reeves gives Abigail a Colt pocket revolver in .22 caliber. Colt made these pistols in calibers .28-.31, and .36, but NOT in .22 caliber.
When Abigail comes riding into the saloon, she ends up firing her pocket Colt 8 times. But pocket Colts only held 5 rounds (some later productions did have cylinders that held up to a 6 shots).
The townsfolk are shown reacting negatively as Passepartout helps Abigail out of the carriage. While interracial romantic relationships were extremely controversial in 1870s America, non of the townspeople would have known at that point about Fix and Passepartout's mutual attraction. They would have likely assumed that the black man helping a white woman out of a carriage was a servant, which is, in fact, what Passepartout is.
Abigail asks Marshal Reeves if his side arm is a Colt Single Action. Reeves says that it is, and adds "Peacemaker". But the setting of the series is 1872 and the Single Action Army did not come out until 1873. Instead, the weapons of choice then would have been either percussion revolvers, or percussion revolvers converted to take fixed ammunition (anything from 1858 Remington Army, Colt Dragoon, 1851 Colt Navy, 1860 Colt Army, 1861 Colt Navy, and many smaller companies that produced percussion revolvers, North and South, during the Civil War).
Bass Reeves refers to leaving Texas for Oklahoma. In 1872, the area now known as Oklahoma was referred to simply as Indian Territory. The Oklahoma Territory didn't exist until 1890.