Post History

Language is an invention much older than civilization.  We have no idea whether all human languages share a single common ancestor language, or whether the capability evolved several times independently.  If there was a single common ancestor, then Chinese, Japanese and Korean are distant relatives[^1].  Distant ones indeed, to the point of non-recognizability.

The Japanese and Korean peoples did initially learn writing from the Chinese, but that didn't bring about any permanent language shift, nor any substantial convergence between the spoken languages involved.  The Chinese writing system was a much better fit for the isolating (classic) Chinese language than for the other two languages, which led to a gradual divergence of the writing systems between Mandarin, Korean, and Japanese.  North Korea recently stopped using Chinese-inspired Hanja altogether, South Korea did so to a significant degree; Japanese Kanji is very prosperous but it is massively dependent on syllabaries added to it inside the unique Japanese writing system.

The ideograms involved are significantly evolving over the 3500 years or so during which they exist; and there's no transnational body that would keep their system and their shapes consistent across the separately managed education systems - so they aren't the same across China, Japan, and Korea, and in fact, not even between PRC and ROC.  

You get a similar picture inside Europe.  Latin-like alphabets are used in much of Europe, including to write Basque, Hungarian, or Maltese, neither of which is Indo-European.  The alphabets aren't identical to each other, but the common inspiration is strikingly obvious.

Every major place in the world uses multiple registers, multiple dialects, multiple languages, often even multiple language families.  These days, you'll find a strong presence of Indo-European languages on several continents not shown on your map; you will also find pockets of Sino-Tibetan languages (including a handful of Chinese language day schools in Japan and Korea) all over the world in constant contact with languages from other language families.  However, this in itself doesn't cause language shift in either direction.

A lot could perhaps be speculated about horses and other military technology, but I'm afraid that it would be off topic for this site.  Military events don't immediately determine subsequent language shifts.[^2]

[^1]: By even mentioning this possibility, I'm violating [an old ban](https://www.frontiersin.org/articles/10.3389/fpsyg.2014.00401/full) on publications that dabble in this notoriously hopeless question.

[^2]: [A historical example](https://en.wikipedia.org/wiki/Mongol_invasions_of_Japan) of conquered Korea and unconquered Japan suggests that horses aren't everything when invading an island and that some Chinese emperors were actually Mongols who chose to learn Chinese as a secondary language rather than impose Mongolian on everybody.