Flag in Around the World in Eighty Days

Sounds like a United States Cavalry guidon:

Cavalry guidons [...] are horizontally divided, scarlet over white, with troop letters and squadron/regimental numbers in white and scarlet, but no branch insignia.

So Wikipedia's example would indicate Troop 1, Squadron C:

And the flag in the movie would indicate Troop 3, Squadron A.

Apparently this design was first used in 1885.

Short answer:

Both the flags carried by the the US cavalry in Around the World in Eighty Days (1956) were inaccurate and/or anachronisms.

Long answer:

The flag with the number and letter is an anachronistic US cavalry guidon, the flag of a company or troop of cavalry.

The number would be the number of the regiment, not the squadron, and the letter the letter of the company or troop. There were no permanent administrative cavalry battalions or squadrons in 1872, though sometimes temporary tactical battalions or squadrons were formed during large campaigns.

Thus the guidon is that of Company A of the Third United States Cavalry Regiment.

In 1876, four years after the date of the novel and the 1956 film - https://en.wikipedia.org/wiki/Around_the_World_in_80_Days_(1956_film) - the 3rd cavalry was stationed in the military department of the Platte which contained Fort Kearny, Nebraska, and took part in the Great Sioux War. And the only transcontinental railroad in 1872 did go by Fort Kearney. So the connection between Fort Kearney, the railroad, Sioux, and the 3rd cavalry is approximately correct.

Actually Fort Kearney had been deactivated by the army in 1871, so Jules Verne probably used a slightly out of date map when writing his novel - a completely up to date map of the US west would have been hard to find in those days of rapid change.

There were three different designs of guidons for the United States mounted regiments - dragoons, mounted riflemen, and cavalry - during the 19th century.

1834 - General Regulations for the Army authorize a red-over-white guidon for companies of dragoons, of which there was a single regiment at the time. It was silk, 27 x 41 inches, with a 15 inch swallowtail, with the letters "U.S." in white on the upper half and the company letter in red on the lower.

1836 - Second Regiment of Dragoons is raised, leading to the issuance of guidons with regimental designations in various formats.

Nevertheless, in . . .

1841 - the new General Regulations for the Army reaffirmed the 1834 design.

1862 - General Order 4 (January 18) directed that "Guidons and camp colors will be made like the United States flag with stars and stripes." The typical design had the stars in two concentric circles with one star in each corner of the canton. Dimensions remained as in 1834. Over the course of the Civil War it became customary to paint the stars in gold instead of silver (which tarnished)

1863 - Army Regulations, Appendix B, directed that the names of battles in which companies had "borne a meritorious part" be painted on the guidons.

1881 - Regulations for the Army of the United States directed that the company letter be inscribed in yellow on one of the white stripes of the guidon.

1885 - General Order 10 returned to the red-over-white guidon, but with the regimental number on the upper half and the letter of the troop (as companies of cavalry were now called) on the lower.

1895 - Regulations for the Army of the United States introduced a bunting guidon for everyday use, reserving the silk for campaign and parade use.

1922 - Change to Army Regulations 129 abolished silk guidons.

1931 - Army Regulation 260-10 reduced the size of guidons to 20 by 27 3/4 inches with a 10 inch swallowtail.

1944 - Army Regulation 260-10 provided for placement of the battalion or squadron number centered in the hoist

http://www.crwflags.com/fotw/flags/us%5Ecav.html1

For some reason actual 19th century guidons often have different proportions than drawings based on their regulation proportions.

https://www.google.com/search?q=us+cavalry+guidon&tbm=isch&tbo=u&source=univ&sa=X&ved=0ahUKEwiZivjitprUAhUDwiYKHYI5B_cQsAQISg&biw=1280&bih=894#imgrc=_2

In the 20th century infantry companies in the US army began carrying guidons, of different design from the cavalry guidons.

Western movies show cavalry carrying guidons of all three designs, as well as guidons that never were carried by the US army.

The United States flag carried by the cavalry in Around the World in Eighty Days should also be questioned. Do you suppose that each fort had several US flags in stock and every detachment that left the fort, of whatever size, would take one along? If so, you are wrong.

Regiments and permanent administrative battalions within regiments carry regimental or battalion flags called colors, or standards in the cavalry. The United States Army follows the 18th century European practice of having two colors or standards - called a stand of colors or standards - per battalion or regiment, unlike many modern armies that have only one color or standard per battalion or regiment.

The two colors or standards are almost always carried or displayed together and separation would be very rare.

They consist of a national color (or standard) based on the United States flag and a regimental color (or standard) based on the full achievement of arms of the United States. The achievement of arms consists of the United States coat of arms on a shield on the breast of a bald eagle and the regimental designation in a scroll.

At the present a regimental color or standard has the coat of arms of the regiment on the breast of the eagle instead of the coat of arms of the United States.

In the Army, most regiments, battalions of regiments, and separate battalions also have a stand of colours. The first is the National Color, which is a 36 in × 48 in version of the national flag trimmed with a 2.5 in wide gold fringe, and is the equivalent of the Queen's Colour in the British Army. The second is the Organizational Colour, which is the equivalent of the Regimental Colour; this is the same dimensions as the National Color, but is of a single colour representing the branch of the service that the unit is from; each branch also has its own fringe colour, which the Organizational Colour is trimmed with. In the centre of the Colour is the eagle from the Great Seal of the United States, but with the regimental coat of arms in the shield. The eagle has in its beak a scroll bearing the regimental motto, with the crest of the regiment's coat of arms above it and the regiment's name below. Attached to the Organizational Color will be the campaign and unit citation streamers awarded to the individual unit – these are equivalent to the battle honours embroidered directly onto the colours of British and Commonwealth units. The Organizational Color was carried in lieu of a National Color until shortly before the Civil War, when the Stars and Stripes became the National Color.3 Civil War era units sometimes carried alternative Organizational Colors based on their home state flags or of other designs.

https://en.wikipedia.org/wiki/Colours,_standards_and_guidons3

The size of colors and standards has also changed. During the Civil war infantry and artillery colors were larger than at present, about six feet by six feet, while cavalry standards were smaller than at present, about 2.5 feet by 3 feet.

US cavalry regimental standards originally had a blue background but it was changed to yellow decades after the Civil War.

https://www.google.com/search?q=US+cavalry+standards+(flags)&source=lnms&tbm=isch&sa=X&ved=0ahUKEwiWi_vpxJrUAhUHJCYKHefGA5QQ_AUICigB&biw=1280&bih=8944

http://www.friendslittlebighorn.com/custerbattlefieldnews.htm5

Also, the US cavalry were not issued national standards based on the US flag until decades after the Civil War and thus after the fictional date of most westerns - including Around the World in Eighty Days (1956).

According to Regulations for the Army of the United States, 1895, Article XXX, Flags, colors, standards, and guidons:

318. The national standard of stars and stripes, as described for flags, will be made of silk, 4 feet fly and 3 feet on the lance, which will be 8 feet 6 inches long, including spear and ferrule. The union to be 22 inches long, and the number and name of regiment embroidered in yellow silk on the center stripe, fringe yellow. The regimental standard will be of yellow silk, same dimensions, the coat of arms of the United States embroidered in silk on the center, beneath the eagle a red scroll, with number and name of regiment embroidered in yellow, fringe yellow,

https://archive.org/stream/regulationsfora00deptgoog/regulationsfora00deptgoog_djvu.txt6

So the US flag carried by cavalry in Around the World in Eighty days (1956) would not be a national standard if it doesn't have a gold fringe around it, if it has normal flag proportions, and if it doesn't have the regimental number and name written on it. It would be just an ordinary US flag - which the army was not authorized to carry - if it doesn't have the special attributes of a national standard.

Thus both the US cavalry flags seen in Around the World in Eighty Days (1956) were inaccurate and/or anachronisms - which is normal in movies featuring the cavalry and Indians.