English and Western are the "standard" styles because they're the ones used in the English-speaking world. Even in the English-speaking world, there are other riding styles, e.g., racing style (i.e., the way that jockeys ride in horseracing).
What defines a riding style? The tack (all the horse gear, such as saddle, bridle, bit, stirrups, etc), and how the rider controls the horse. These are not independent - the way that a horse can be controlled depends on the tack, and the tack has evolved towards a style of controlling the horse. English and Western are distinguished by different styles of saddles, and differences in how the horse is controlled. If you look at horseracing tack and the way that jockeys ride, it's clearly yet another different style.
Even in Europe, there are different styles. For example, there were three old riding styles in Spain: a straight-legged style, "a la brida", being similar to other western European styles, and a strong influence on Western style riding (due to the Spanish colonial presence in America), a bent-legged style, "a la jineta", derived from Arab and North African styles (and still used by riders in bullfighting), and an intermediate style).
Also, in the Americas there is a variety of styles. Chilean style is similar to Western, but uses a different style of saddle. Plains Indian styles were also different, without saddle or stirrups.
So, unsurprisingly, Asian riding styles are also diverse. Many of them are derived from a common steppe nomad tradition, and tack shares common elements. However, there are regional differences in both tack and style, and together these can result in differences between various Asian styles greater than that between Western and English. Further, there are many and important differences between these styles in general and Western/English. A good illustration of these differences can be seen in these instructions to English-speaking tourists for equestrian tours in Mongolia: http://www.mongolia-travel-and-tours.com/horse-riding-mongolia.html - some of these instructions make more sense once you see what a traditional Mongolian saddle looks like:
The high front and back allow the rider to control their horse by shifting their body weight backwards and forward. To stop, shift weight back. A Western/English rider can quickly run into trouble if they can't stop their horse! Also, frustration (but less danger) if they can't start their horse.
Saddles also serve to illustrate some of the differences between Asian riding styles. This Manchu-style saddle (Qing Chinese):
is clearly similar to the Mongolian saddle, but less severe. Other related saddles are:
Ming Chinese: https://commons.wikimedia.org/wiki/File:Ming_saddle_(Great_Wall_Museum).jpg
Tibetan: https://www.metmuseum.org/art/collection/search/35375 - closer to the Ming saddle than the Mongolian saddle
Japanese: https://commons.wikimedia.org/wiki/File:Left_side_view_of_Kagami_gura,_Musashi_Mitake_Shrine.jpg
Kyrgyz: https://commons.wikimedia.org/wiki/File:Saddle_Gear_of_Central_Asians._Karakyrgyz_Saddle_Gear_WDL10874.png - similar to the Tibetan and Ming
Tunisian: https://commons.wikimedia.org/wiki/File:Horse_saddle_Sahara_Festival.jpg
However, since most of the riding styles in this great variety don't appear in equestrian competition in the English-speaking world, they are little known there.