I'm going to preface this by saying I am not an expert in Mongol/Mongolian studies nor in the history of education in Japan. Others may be more qualified to comment. However, from what I can gather, the overarching answer is that the vast, vast majority of Japanese in the 19th and early 20th centuries knew nothing about the Mongols. As I understand it, "history" as a topic was not really taught prior to the education reforms that took place after the Meiji restoration. Schools, themselves, weren't even "common" in the same way that we think of modern schooling. Temple schools (terakoya, 寺子屋) that taught reading, writing, math, etc didn't really pop up until the Edo period (17th-19th centuries), and prior to that, education was generally reserved for nobles, samurai, and clergy with an education focused on reading, writing, and arithmetic (naturally), along with the Chinese classics such as the Laozi, I Ching, etc. Such influence can be seen in various cultural aspects of the time, from incorporation into Court ritual to concepts in martial art schools and more.
Generally speaking, I'm not sure history was taught or successfully taught anywhere in the world before the modern era to commoners. How much of ancient Greek/Roman life did a French commoner or noble know in the 18th century outside of the classics? China would be a good bet for more structured historical learning, as there were numerous institutions across different dynasties for educational awareness. But history as a subject is odd because it requires a degree of objectivity in order to know and understand. We're lucky in this forum to be able to have such objectivity. A daimyo in the Edo era with a less-than-favorable view of the Tokugawa during the Sengoku Jidai during the 16th century likely would not be able to get away with that. Similarly, Chinese history of the Mongols at the time of their existence (13th century) were far from objective. As such, history for commoners was found more in stories and the classics than any real legitimate scholarship... and only if the subject was a common one.
Turning to the Mongols, there was no successful invasion of Japan, which meant no real impact to daily life that would be cause for historical remembrance. After the failed invasions, warriors that participated in the brief fights in the Hakata/Kyushu area did submit demands to the Court for rewards/title; in 1275AD, the bakufu rewarded 120 warriors for their services, for example. Success in routing the second invasion that never made it to the mainland yielded a similar result. Shortly after the second attack, there was intense factional conflicts at Court and it wouldn't be long until the Kamakura bakufu fell. Oral history of family title/reward due to Mongol invasion attempts may have been remembered, but scholars didn't suddenly pen the great 元軍記 (Kengun-ki, "Story of the Mongol Forces"... that's what I suspect it would have been called if it existed). Changes emerged in the Kyushu region as a result of the failed invasions, though they were not of real impact to daily life and, thus, not historically captured for posterity in any great degree.
What we do know is that Takezaki Suenaga, a warrior during the invasions, commissioned two sets of handscrolls in the late 13th to early 14th century. These scrolls called the moko shurai ekotoba 蒙古襲来絵詞 and, apparently, they were only known to a few warrior families in Kyushu for ~500 years before they were widely disseminated across Japan. The scrolls were considered valuable enough to not be discarded, but they were not given any special care. Historians also note that the scrolls were altered over time. From what I understand, Suenaga's name was even scratched out by another family in the 16th century (the Oyano 大矢野). It wasn't until Arai Hakuseki, a scholar/advisor to the Tokugawa bakufu in the 18th century, that these scrolls came to be seen as a historical source.
Fun fact - Suenaga's family suffered during the transition to the Muromachi bakufu and the Nawa family 名和 ended up with the scrolls until Nawa Akinori's marriage into the Oyano family. What makes this fun is the Nawa family's association with warrior groups and shrines at the time, including those who may be considered ninja. Kinda cool to think ninja were keeping such secrets in the 15th century!
Early histories of the invasions in Japan rely almost exclusively on the Yuan-shi 元史 (History of the Yuan Dynasty) from China, compiled by the Ming after the Yuan dynasty fell. The Isho Nihon-den 異稱日本伝 (Treatises on Japan under Different Titles), written in 1537 and published in fifteen volumes in 1693, contains references to the dynastic histories written in China and Korea (the Yuan-shi and the Goryeosa 高麗史, History of the Goryeo Period, both of which were compiled approximately a century after the invasions). The Gyoju Gaigen 馭戍概言 (General Remarks on Mounted Defenses), written in 1778 by the Kokugaku 国学 scholar Motoori Norinaga, likewise refers almost exclusively to the Yuan-shi in reconstructing the Mongol invasions of Japan. Note that the Yuan-shi was not a commonly read or even owned source in 19th and 20th century Japan, let alone prior to the Meiji restoration.
To add to the complexity of the topic, the Yuan-shi was a "rush job" in China, being compiled in less than a year. Apparently it was criticized heavily for its mistakes. The section on Mongols was also lacking, due in part to the hatred felt towards them by the Ming compilers. Mongols were described as barbaric and ruthless, important aspects of the culture ignored and left out due to ignorance or choice (noticed by later scholarship). Translations from Mongol sources supposedly were not good. On top of that, compilation of the Yuan-shi took place in Nanjing whereas remnants of the Yuan archives at the time were in Beijing 600+ miles away. Not the best work to be done by court compilers.
All that to say knowledge of the Mongols in 19th and early 20th century Japan was most likely confined to what the Chinese wrote in the Yuan-shi combined with biased/altered pictorial evidence from the moko shurai ekotoba (eg, a whole new scene with Japanese carrying mongol heads was added in the 19th century in one reproduced set!). And that knowledge was most likely confined to a super small percentage of elites who purposefully looked for that knowledge... and it was not very good/accurate.
Best I can offer, sorry it's not as thorough as it probably should be. It taught me something new trying to put an answer together!
Sources-
Supposedly Japan in the Chinese Dynastic Histories by Ryusaku Tsunoda is a good source for the Yuan-shi in English. I don't own a copy and my nearby library doesn't have it to verify :(