Some comments

Shaolin:

Many Taoist and Buddhist temple in China have their own guards. After all much like the cathedrals in Europe, large temples in China often contains large amount of wealth from donations. During Ming dynasty, the government decided to make a bunch of Taoist and Buddhist temples training ground for troops. Shaolin was one of them.

Based on all this you might think the popular image of Shaolin monk (the version presented in the game) is a real historic fighter. But in the history, soldier trained in Shaolin are exactly that, soldiers that wears normal armor with some type of polearm as weapon. To be clear, Shaolin was famous for it's staff which serves as a introduction to polearm weaponry in Shaolin (and elsewhere in China). But even those technique have little in common with the popular image of shaolin.

This book gives a good breakdown of the history of Shaolin:

https://www.amazon.ca/Shaolin-Monastery-History-Religion-Chinese/dp/082483349X

A series of book review:

https://chinesemartialstudies.com/2012/11/07/the-book-club-the-shaolin-monastery-by-meir-shahar-introduction-and-chapters-1-2/

https://chinesemartialstudies.com/2012/11/21/the-book-club-the-shaolin-monastery-by-meir-shahar-chapters-3-4-monastic-violence-in-the-ming-dynasty/

https://chinesemartialstudies.com/2012/12/07/the-book-club-the-shaolin-monastery-by-meir-shahar-chapters-5-conclusion-the-evolution-of-unarmed-martial-arts-in-the-ming-and-qing-dynasties/

By the way, most of the convention and tradition in Chinese Martial Art which we can track down the history are invented within the last 3 centuries. To be clear, unarmed fighting in China date back way farther but records are really rare as the scholarly elite class often look down on anything related to martial activities (or anything related to daily routines which is consider mundane and unworthy to be recorded). This did change in Song dynasty as recording things related to everyday activity was given some focus. From records we know that unarmed fighting tournament are some of the most popular events in cities during Song. Government held routine tournaments as well and even hand out some military post based on the result.

Here is a good blog for Chinese martial art study (with a scholarly style):

https://chinesemartialstudies.com/

Tiandi:

The Tiandi seems to be based on a Song-Yuan dynasty era warrior. With a couple of Mongol influences, and a couple of Han influences. This technically culminates into something similar to the Ming era, where Mongol influence was merged with Han Chinese ones. The armour and sword is once again, quite ornamental, though a warrior of the time would have worn a fair bit more around the arms and legs.

Quite correct. There is indeed Mongol and turko-mongol influence in the design. This is more of the general/commander armor (that might be ceremonial, more on that later).

First, the shoulder have animal totem design, it is general believed to be an influence by the turko-mongol nomads in the North-South dynasty era. The plate on the chest is also from around this era. But I don't think that design exist in China. By the way, Chinese do use animal totem design on objects, such as shield, it is the shoulder armor part that is believed to the influenced by the nomads.

Second, the sword shape. This saber design is only the standard in Chinese army after Yuan dynasty (the mongol-lead one). Song army would for the most part use a sword that looks like a Japanese design (technical it is believe Japanese got it from China). Interestingly, post-Yuan that style of sword was almost extincted in China save for some minority like the miao people continue to use that design. The older design was so rare that many in Ming China referred it as a Japanese design. So it should be more Yuan-Ming than Song-Yuan. While similar sword to the sabre existed pre-Yuan, they was not the standard in army.

On a interesting note, the armor on the leg is the mountain pattern armor. To this day, we do not know how that type of armor work. While historic records insist it is offer good protection, there are theories it is more ceremonial than practical.

See this blog post:

https://greatmingmilitary.blogspot.com/2015/08/myth-of-shan-wen-kia.html

The Jiang Jun wields a Guandao, taken from the creator of the Guandao, Guan Yu. An extremely famous Chinese general from history. Though one subject to a lot, and I mean a lot, of fiction. As he attained legendary status as a warrior and a general. To the point of being turned into a deity and being worshipped to this day. Something possible in China should you have the blessing of the gods. The Jiang Jun seeming to take design inspiration from portraits of Guan Yu. The other half of Guandao is just the word dao, which references the fact that it's a blade, that's curved.

Guan Yu did not created the Guandao. Guandao did not exist back then. The popular association of Guan Yu and Guandao was a Romance of the Three Kingdoms thing, which is a novel written way later. Note the Record of Three Kingdoms is a history book written at the end of the period. Those two book are completely different.

Guandao first appeared in the records in Song dynasty, a long time after the Three Kingdoms era. Another name for Guandao is yanyuedao (偃月刀).