It's time to mask one's ignorance with the help of Wikipedia!
I'm no expert on architecture, so this part is mostly about knowledge that are on the level of common sense. There are of course, more architectural inaccuracies as severe as AoE2's Asia style in our panhistorical games.
(AoE2 Byzantine's Islamic style has already been mentioned. How about AoE3 putting a Catholic cathedral in Amsterdam? Or Civ5's background statue for its American-accented Chow Yun Fat-lookalike Ramkhamhaeng...)
An RTS that gets East Asia architecture?
Someone said that the only games to get the art style difference between China and Japan right is Liquid Entertainment's Battle Realms. My memory of the game is hazy, so I cannot examine the designs of its buildings rigorously, but his general point stands: the art design of Western-desveloped games often doesn't understand East Asia architecture enough to differentiate between China and Japan. Here, the art direction in the RTS branch of the Civ-AoE-RoN lineage will be put to test.
Age of Empires
The development of AoE1(working title "Dawn of Man") was a complicated process, as Ensemble kept trying out different new ideas, until they finally decided to model the gameplay after Warcraft II. However, even then the game wasn't meant to include any East Asian civilization; they only decided to add East Asian civs when they realized that Asia can be a market for RTS games.
In AoE1, Japan and Korea are represented in AoE1 as Yamato and Choson. The name "Yamato" appears in history far later than AoE1's general time period, and Ensemble's decision to make a Yamato campaign resulted in the campaign's final mission being a alternate history scenario against China's Tang Dynasty, which belongs entirely in the time period of AoE2. China is curiously represented by Shang Dynasty, which is designed as a beginner's civ (i.e. overpowered and often banned) with reduced Villager cost, and has no access to Iron Age techs in the Government Center building. The Japanese version of AoE1 renamed Shang into Han (the ethnic majority group of Chinese).
When it comes to architecture, AoE1's Asian style is a very generic one. The blue color of the roofs makes one suspect that these roofs are covered in glazed tiles, which would be expensive at any point in history. The Asian wonder is modeled after the Temple of Heaven, which is built in 1420. Of course, we would see the Hall of Prayer for Good Harvests' unique round figure again and again, in both AoE2 and AoE3:TAD.
Originally, the Asian civs' Bronze Age Market and Town Center used torii gates (screenshots at the AoEBeta site). Ensemble changed those when they realized that torii is strictly for religious purposes. But AoE2 would go on to prove that they hardly learnt any real lessons about East Asian religion or architecture.
Age of Empires II: The Age of Kings
In AoE2:AoK, the East Asia civs are Chinese, Japanese and Mongols. However, with a few exceptions, Ensemble apparently used pictures of Shinto shrines exclusively as their reference for Asian architecture. The results, beautiful yet very wrong, however many years after the game's release, still look hilarious.
Basically, AoE2's Asian Feudal Age style is based on a very specific thatched-roof shrine style, and the Castle Age buildings (except for Town Center) are based on another specific style whose roofs are in straight lines, without curves. The Shinto shrine styles are very distinctive, completely different from buildings commonly found in East Asia (which are mostly developed from Tang architecture). The fork-like chigi structures on roofs are one of their most signature features, and never appears on anything else, yet all these AoE2 buildings have chigi on them. If Civ4's Palace design is based on Shinto shrines (I'm not sure, but the structures on its roof look like chigi), that would have been wrong, too.
AoE2's Asian Monastery has a large bronze bell, suggesting it to be a Buddhist temple. The Monastery also has a torii gate. The torii gate does appear in Japanese Buddhist temples, but the problem is as an iconic symbol, it's mostly associated with Shintoism, so the situation is similar to AoE3 Germans building Catholic Churches: it's not that Germany has no Catholic churches, but that for a German, an archetypical church is a Protestant one. Needless to say, the torii is also Japan-only.
The Asian Castle is clearly based on the tenshukaku keeps of Japanese castles. The Asian Imperial Age buildings switch to a more generic style (however, in AoE2, only Town Center, Market, University and Towers have Imperial Age forms). The Imperial Age Market has Chinese features such as the duilian at its entrance and the facade of its main hall.
To be fair, Ensemble was only trying to find 2 highly recognizeable styles to differentiate between Feudal Age and Castle Age, but their choice ended up wrong on a far larger scale than AoE1's torii gates.
Rise of Nations
In comparison, the RoN artist team had clearly done more work on East Asian architecture. For example, the Asian Ancient Era University has an observatory, whose shape is modeled after the Korean Star-Gazing Tower / Tower of Full Moon.
The Koreans are the only nation in core RoN to have its own unique architecture style. The roofs are green, again suggesting glazed tiles, and the curve of the roof goes upwards in a way peculiar to Korea architecture. (Buildings of Southern China also has upward-curved roofs, but that's another different style.)
RoN also has entirely made-up buildings. The Asian Ancient Era Temple is white in color, reminiscent of the Tibetan Buddhism style of stupa, but the design is something not seen in real life, attaching the octagonal top of a pagoda to a rectangular ground floor. In contrast, RoN's other fictional architectures, such as the European Ancient Era Temple and University, which combined Stonehenge and Greek pillars into one invented style, are more interesting.
Empires: Dawn of the Modern World
In this criminally overlooked game by Stainless Steel, the playable Asian civilizations are the Chinese and Koreans. The game's art direction means the player's base never feels like a city, and always resembles some small villages, but at least its Asian architecture does look like real Chinese / Korean villages, except with a few strange visual easter eggs thrown in such as "home sweet home" written on a flag.
Empire Earth II
In this game, the only buildings that reflects a nation's culture style are the Town Center and the wonders, so there isn't much to note, except the Ancient Wonder of the Far Eastern region is "Tower of the Moon and Stars", which is the aforementioned Korean observatories conflated into one building.
Age of Empires III: The Asian Dynasties
Developed by Big Huge Games, this time they put the wonder of their own choice, the Porcelain Tower of Nanjing, first seen in Rise of Nations, into the game. Since Civ5 also featured the Porcelain Tower, perhaps Big Huge Games has successfully left their mark on strategy games by popularizing it, just like Ensemble has left theirs by popularizing Chu-Ko-Nu (repeating crossbow) and Trebuchet.
- The lineup of playable nations in the Civilization series, Addenum 2: East Asian architecture