Early on, Graham notes with surprise that Algren is somewhat of a scholar and Bagley comments that Algren can speak Blackfoot. His ability and willingness to learn languages helps him learn some rudimentary Japanese within the few months of his imprisonment.
It works well also in that the Japanese language, as far as syntax and structure goes, is remarkably easy to learn in comparison to many other languages. It's learning the words for nouns and verbs and such themselves that tends to be the hard part, but once you have those down to a decent degree, it's a simple matter to start stringing them together into sentences.
The samurai way of life and that of the Native Americans that Algren fought against are very similar. Both are spiritual people who strive for living in harmony with nature and both groups are being pressured by a more modern, invasive force that is not afraid to use violence to get what it wants. It's not hard to see why Algren bonded so quickly with the samurai and fell in love with their culture.
Also, since Algren helped destroy the very people that he respected, which led to his mental breakdown and alcoholism, fighting with the samurai could be seen as a way of him atoning for the atrocities that he helped commit.
Katsumoto's reverence for Custer's Last Stand makes sense when you realize he never intended to win in the first place.
The first bokken fight between Ujio and Algren is representative of the overall problem that the samurai have. Algren gets his ass kicked, but won't give up and keeps trying. Later, the samurai realise that even if they defeat the Imperial Japanese Army, the government will not accept defeat and simply conscript more troops for their regiments, while the ranks of the samurai will dwindle even in victory.
When Algren was fighting off the four samurai during the fog battle, notice how the spear he was using to fend off the four samurai has a white tiger; according to Chinese mythology, the white tiger Bái Hu (known as Byakko in Japanese) is representative of the West/westernly winds. The white tiger is also connected with the element of metal and is represented as a symbol of strength, courage, willpower, and war. It was also a common symbol used to represent military officers, at least in the case of Korea.
With this in mind, one could say that Algren was the living representation of the white tiger on the banner attached to the spear he was using: he is a white (former) US Army/Cavalry captain who came from the United States (a Western nation that just relatively recently concluded the Civil War and was a relatively modernized nation that possessed a formidable military) to help train and eventually lead a newly-formed Imperial Japanese Army (historically inaccurate, but shhh..) in order to crush a samurai rebellion. During the fight, Algren fought like a cornered animal, displaying strong willpower and ferocity (then again, adrenaline and a primal survival instinct does have a rather dramatic effect on you when you’re cornered and have no other options left).
There's also probably some Rule of Symbolism invoked for Dramatic Irony here: while the Satsuma Rebellion / Seinan War was the primary inspiration for the film, an earlier conflict did involve a proud samurai domain (fighting with anachronistic weapons) losing a long battle against a better-equipped Imperial army. That happened to be the Battle of Aizu, and one particularly tragic episode involve a detachment of samurai committing suicide. Their name happened to be the "Byakkotai" (White Tiger Corps).
Nobutada losing his topknot is a callback to an earlier scene when Simon Graham trips Algren's PTSD by expressing his "dread fascination" with scalping. The way the scene is shot and blocked, all that's needed is the knife to start cutting a little lower. Notice how Graham is front row center to see the act and looks like he's about to lose his lunch.
was the primary inspiration for the film, an earlier conflict did involve a proud samurai domain (fighting with anachronistic weapons) losing a long battle against a better-equipped Imperial army. That happened to be the Battle of Aizu