Though it is easy to see such a Korea quickly establishing itself as Empire earlier like OTL Japan and fighting off the latter's attempt to take over, would it have been possible for both post-Meiji Japan and post-Munjo Korea to become the closest thing to a far eastern version of Austria-Hungary (albeit with two region-specific co-Emperors) instead of one dominating the other?
Austria-Hungary was a very special case where the German plurality was only slightly (24% vs 20% in 1910) larger than the next largest group, the Hungarians, and the two composite kingdoms were nearly equal in land area. The Austrians had to concede to the Hungarians and give them the Ausgleich after the Empire was wobbling from its defeat by the Prussians and it needed the support of the Hungarian upper class to keep the nation from falling apart financially and cartographically. Not only that, the Austrian Empire was composed of a great many ethnic and cultural groups, of which the Austrians and Hungarians only made up a plurality and compromise between the two was necessary for managing the nascent nationalistic movements springing up all over Europe in the wake of the chaos of the mid-1800s.
A Japanese-Korean entity lacks quite a bit of those same pressures. The Japanese population has historically consistently been larger than the Korean population, reaching 3x of the Joseon population during the Imperial era, and Japan proper+Taiwan is 2x the size of the Korean peninsula. Including Manchuria adds a huge number of Chinese into the mix but Japanese still remains the majority of the Empire in total and integrating the Chinese into a foreign nation when a Chinese nation exists is a bit...problematic? Anyways, even if the Koreans conquer Manchuria first in Munjo's time, it wouldn't really an integral part of their Empire the way the Carpathians and the Balkans were for Hungary due to length of time it's been held, so it'd make no difference to the Japanese to just administer it themselves versus having the Koreans do it.
In the end, the Japanese would inevitably dominate that sort of union just from sheer population and land area if it starts out equal.
In terms of an alliance, that too is a bit problematic since the Japanese were very sensitive about who held the Korean peninsula (a dagger pointed at the heart of Japan) the way the British were about Antwerp (a great fear of a foreign power making use of that staging point to invade the country which never actually paid off because no one has ever used that staging point and any attacks or invasions planned involved different ports and routes) and there was plenty of propaganda and influences to invade the Korean peninsula a third (or fourth, if pre-1000 stories are to be believed) time (Saigō Takamori wanted it in the 1870s) while the Joseon historically looked down on the Japanese as an inferior, upstart culture. The Joseon would have to modernize and remove tributary relations with the Qing before Japan can modernize (which is a difficult task in and of itself due to the plethora of conditions which made Japan's Meiji Restoration and its rapid social, economic, and political change possible, like the trade with the Dutch that the Koreans did not have and Western Great Power attention that focused mainly on the vast untapped markets of the Qing and Japan) so that there is no weakness to be found and exploited by Japan. Issue is, the Joseon didn't have military, political, or economic incentive to fight the Qing because they'd tried twice before (back when the Qing only had Manchuria with terrible results) or to modernize and self-strengthen until it became apparent that the Qing could no longer protect them, which was after the Europeans and Japan beat on it, which provides a point which is far too late to catch up with Japan. An early reformer would, unless the circumstances made it absolutely necessary, be deposed, poisoned, made ineffectual, or just removed like the number of Joseon kings with unorthodox opinions who tried to institute policies conflicting with traditional Confucian norms and the court's own desires (like King Gwanghaegun was exiled for trying to avoid war with the Manchu, Prince Sohyeon died under mysterious circumstances maybe involving his own father and an ink slab to the head, Jeongjo likewise killed, and all the other political nonsense that plagued the Joseon).
I'd say that the Joseon would have to have suffered a different Imjin War, one that confirmed the need to modernize even more due to the unreliability of Chinese aid (to force independence and remove all complacency) and centralized power from the ministers to the king so as to avoid the court politics that deposed kings and allowed yangban families like the Andong Kims from putting their puppet child kings
on the throne. Post-Imjin War, the nation's devastated, tied down to the Ming (which the court would use to drag the Joseon into conflict with the nascent Manchu), and in the control of the yangban, who end up deposing Gwanghaegun and Injo, revealing the complete lack of royal authority in the kingdom, and stagnating the country for another 300 years.