As a native Chinese speaker, 勉强 means something very different (it means to be reluctant or forced to do something) for the same kanji/漢字. Might anyone know how it came to mean learning or studying in Japanese?
According to Morohashi (大漢和辞典), the original Literary Sinitic meaning of the compound word, 勉強 [m-jenX g-jangX] are indeed different: "1. Strive with all effort. 2. Put effort in study. 3. Do business with small profit margin. 4. Section from the book 顔氏家訓. 5. (modern Mandarin) force; compel."
In fact, definitions like this are closer to the modern Japanese meaning (to study and even to discount) than to the Chinese ones (to force, reluctantly, though barely is similar to Meaning 1.) But we can do better than that and search for the evolution of the Japanese meaning.
According to Nihon Kokugo Daijiten, the first usages of the word in native Japanese texts used it in the meaning similar to "[person] surpassing difficulties with great effort, enthusiastic, zealous", attested at least from 17th Century. A century later, it evolves to "reluctant, unwilling" (putting effort to be unproductive), and even later, in Meiji era, gains a connotation of "studying and striving all the practical skills to be modernized and European". "Discounting" also appears around the late 1800s. Neither of these is extremely away from the Classical meanings; however, the emphasis on study seems to be an explicit Meiji thing.
Thus, probably, a real question here is "why did the Chinese meaning drift so hard?" The destination is still easy to see: the shift "Put effort into something > put effort into making someone do something" is not exceptional.
To conclude, all the meanings are not so different and surround the same semantic fields. "Reluctant", not mentioned by 大漢和辞典, but becoming the only meaning of 勉強 that appears in Korean and Vientamese, to say nothing about appearing in Japanese as well, is a bit aberrant, but as an ironic extension understandable (and Hanyu Dacidian has a quotation with that mweaning from Records of the Three Kingdoms, so we cannot claim it's new!). The shift to "study" in Japanese is explicable by Meiji realities, and the remaining usages both in Chinese and Japanese are allowed by Literary Sinitic. The modern Mandarin usage "to compel" (also in Cantonese) is a bit strange, but also understandable.