First of all and most importantly, as said in the main article, the setting for this film conflates the 1868-1869 Boshin War and the 1877 Satsuma Rebellion in a single conflict that takes place in 1876. From there...
Shockingly, Algren is allowed to wear his cavalry saber in front of the Emperor, when in real life he would have been forced to leave it outside the chambers.
Near the start of the film, General Hasegawa loses the battle with Katsumoto's samurai and commits seppuku, with Katsumoto himself acting as the kaishakunin (someone who beheads the person committing seppuku, so they don't die a prolonged agonising death). The movie, however, gets several elements of the process wrong:
Katsumoto severs the head clean off, instead of leaving a small part of the neck attached so that the head doesn't roll away (which it does). This might be chalked up to it not being the best moment and place, with Katsumoto being presumably all tired and worn after the battle, but it's still jarring that a skilled swordsman with a strong grasp of his traditions would not be able to honor his role in the ceremony.
Katsumoto precedes his strike with a screaming cry, which is completely against the role of the kaishakunin and the ceremony of seppuku altogether. Seppuku is meant to be a solemn moment of ephemeral dignity, with all focus being directed on the poor guy carving his guts open. Screaming not only redirects attention to the kaishakunin, it also shatters the calmness of the moment.
Because Hasegawa was an enemy general, it is very unlikely he would be trusted with an actual blade (because, "rush at your opponents with the weapon at hand and die trying to take as many of them down with you as possible" would have been seen as just as honourable as death via seppuku). Most of the time when an enemy combatant was given the opportunity to commit seppuku, some facsimile (typically a fan) was used instead of a knife, and the kaishakunin would make their cut as soon as the person committing seppuku reached for it. This is perhaps slightly more forgivable in The Last Samurai, as it's stated that Katsumoto and Hasegawa had previously fought together, so it is possible that Katsumoto trusted the general not to betray him... though if that level of camaraderie existed between them, it raises another problem of why Katsumoto let Hasegawa make the initial cut in the first place (the kaishakunin could make their cut whenever they felt the ritual had gone on long enough; if they liked you, they would do it before you even made the cut across your abdomen, so as to spare you the pain of the cut and the risk of crying out in agony, which was seen as shameful).
Contrary to popular belief and what the film claims, the concept of bushido is not ancient or traditional at all. While the term has always existed in Japanese language (it just means "warrior's way"), its meaning as we know it today, that of a sort of unified ethical code widely accepted and cultivated by the samurai class, was largely popularized in the 1900s during the Meiji period, when the government of Japan turned to the past in the search of spiritual inspiration in midst of its process of modernization, leading to the invention of a romantized, supposedly traditional chivalry code that emphasized exactly the values the Meiji period Japan was trying to instigate (in this case, the selfless devotion from an Imperial soldier or sailor to the Emperor). Prior to that, the philosophical codes upheld by the samurai were heavily dependent on the era, the clan and the individual in question, only with some common influences from Zen Buddhism, Shinto and Confucianism that might or might not coincide in their interpretation.
Even if we take the bushido from the film as just a naming convention for period-accurate samurai values, those are not even portrayed very faithfully either — the film generalizes the most zealous historical elements of the samurai culture to the whole of them. For instance, the extreme emphasis given to loyalty to one's lord, to the point of being willing to take one's own life with just an order, was never that common among historical samurai, many of which often turned their coat during battles at the first impression they would be on the losing side. While dying on your lord's service was always considered honorable and fit for a warrior, actively getting yourself killed in a pointless way, nor matter how honorably or faithfully, was something only popular among a minority of really idealistic elements (such as Yamamoto Tsunetomo, who famously stated in The Hagakure the samurai's purpose was found in death — needlessly to say, many would have disagreed). The same goes about unwinnable battles, as although samurai choosing to fight to the end were lauded, doing the smart thing and retreating wasn't disgraceful either.
The plot point of Algren shedding his self-destructive tendencies thanks to the samurai ethos is by itself an amusing irony. As said in The Hagakure, the ideal samurai is willing to die a worthy death for one's lord and duty, and finding any external reason to avoid this would mean moving away from this purpose. Therefore, the instance where Algren resembles an ideal samurai the most is at the beginning of the film, when he's basically a soldier that does not fear death and only comes to do his job for the U.S. Army, not when he outgrows this trait and finds a reason to live with Taka. There is a slight subversion in his last lines, when he pledges his loyalty to the the Emperor and promises to commit seppuku himself at his order (although, as said above, most samurai would have found this an exaggeration), but this offer is implied to be only to show how strongly he has adopted Katsumoto's ways and on the natural assumption the Emperor will not give such an order.
The notion of the ancient samurai as "protectors of Japan", which Katsumoto laments he has failed to live up to, is a complete fabrication. Not even at their most idealistic would historical samurai have claimed such a thing, much less expect it to become some sort of nationalist ideal for their followers. Samurai were first and foremost vassals and members of aristocratic families, and their interests were always tied to those, which prevailed even in a unified Japan. The only points of history where samurai fought to protect Japan in any capacity were those where most of Japan had joined forces against a common invader, such as the Mongol invasions (those depicted in Ghost of Tsushima), but those happened in a pre-unified Japan, and thus couldn't be attributed any nationalistic fervor beyond the circumstances. In the 1870s, when the movie is set, glorification of the samurai themselves would have been seen as glorifying a backwards, even barbaric past.
Warfare
At the start of the movie, the Imperial Japanese Army is depicted as an undisciplined, poorly trained mob, completely unprepared for any major military action despite all their modern guns and uniforms. In contrast, the samurai rebels are portrayed as fearsome lifelong warriors that only know defeat through heavy firepower. This is pretty much bizarro world: in real life, Japan already had a well-armed, modern and professional army by the 1870s, which had seen plenty of combat experience in the Boshin War, as well as in the 1874 Japanese invasion of Taiwan and the Ganghwa Island incident of 1875, while in turn, most of the samurai rebellions that took place after the Meiji Restoration were in fact undisciplined mobs with various levels of competence. Tokyo would suppress them fairly quickly, helped by the fact that those revolts were usually highly localized.
Japan did not use American advisors for military matters; their Army used French (before 1870) and later German advisors, while their Navy used British advisors. Most American advisors in Meiji-era Japan were educators and agronomists.
Although some samurai did consider guns to be dishonorable, nearly all of them recognized their necessity in modern warfare since long before the Edo period. Actual real life examples of samurai lords who had the opportunity to equip their army with guns but made a deliberate choice not to are very hard to find. During the late Sengoku era, firearms either imported from the Portuguese Empire or patterned after Portuguese arquebuses proved decisive in many battles, such as in the Battle of Nagashino in 1575, where the combined forces of Oda Nobunaga and Tokugawa Ieyasu routed the Takeda clan largely thanks to Nobunaga's use of volley fire against the Takeda cavalry.
As an extra point for the previous, during the Bakumatsu period, forces loyal to and opposing the Bakufu sought to acquire as much western military equipment as possible. The samurai who took part in the Satsuma Rebellion, which this movie is based on, did indeed use rifles and cannons. They would also use traditional weapons like swords and bows, but this was mainly as a result of running out of firearms ammunition.
In line with what is said above, most combatants in those conflicts wore the modern, western-style uniforms that were used in Japanese armies at the time, while in the film, the rebellious forces are all clad in ancient samurai armor. This kind of armor had been abandoned many years before because it couldn't stop bullets and was too heavy and cumbersome in turn. Wearing one on the battlefield, even in order to make a point of traditionalism, would have been a very eccentric act that only a commander could afford to do; the man Katsumoto is a stand-in of, Saigo Takamori, really did so, but only in his very last battle.
The film features ninjas, who, as typical in Hollywood media, are portrayed as acrobats in black suits who appear to kill their target with exotic weapons and in spectacular fashion. Real ninja didn't use such clothes or tactics. They would have simply disguised themselves as villagers and found some discreet occasion to assassinate Katsumoto, maybe with poison or a stab in the dark, but certainly not in the open or surrounded by an entire crowd.
Characters
The Emperor is presented as the Emperor Meiji in his introductory scene, which is a mistake. In real life, Emperor Meiji was actually his posthumous title, a name referring to his role as the emperor of the Meiji era, which wasn't used while he was alive. His personal name was Mutsuhito, but at the time he would have been officially referred to exclusively as "The Emperor".
Also, the Emperor is described and accordingly portrayed as fickle and weak-minded, which he wasn't in real life at all. Indeed, transforming Japan from its previously isolationist, feudal state to the industrialized world power it turned into would have simply not worked with the kind of borderline Puppet King shown in the film.
