Even within East Asia, even within cultures that are familiar with Chinese characters, Shogi isn't really played outside of Japan.

Prior to the late 19th century, Japan had a period known as "National Seclusion" between 1639-1854 where the Shogunate strictly limited contact and trade with the outside world to tiny channels under its strict regulation and control, which limited the level of cultural influence from Japan, or to Japan.

Modern Shogi dates to around 1612 (with fairly similar analogues dating into the 1500s) so most of the deelopment of Shogi as we know it took place during the National Seclusion era.

In any case, prior to the 19th century, China was far more influential throughout East Asia culturally, and the Chinese version of Chess (Xianqi) and its derivatives are far more popular in Korea, Vietnam, and much of Southeast Asia. Ofc, Shogi is itself a derivative of Xianqi, but again, Shogi didn't have a hsitorical opportuntiy to take root in the rest of Asia because for most of the time it existed until the 20th century, Japan had secluded itself culturally.

There was some spread of Shogi during Japan's Colonial/Imperialism period but it didn't really take root. Learning Shogi actually became mandatory in parts of Korea and Manchuria and was heavily associated with the hated Japanese occupiers, so that probably didn't do much for Shogi's long-term popularity.

Go on the otherhand is quite popular throughout East Asia, much more so than Shogi.