Was it a question of subject matter, organization, affiliation, or other factors? Was the establishment of a Western-style university a major or minor adjustment for China as a whole? And why establish new institutions rather than reform existing ones?
I've noticed similar dynamics at play in other Asian countries (e.g. Thailand and Japan), where education had a strong tradition and legacy but the "first university" dates to no earlier than the late 19th century, so info from Asian countries outside China would also be appreciated.
I can discuss the history of Japanese Educational systems, and so in this response I will.
As you know, education didn't start with the adoption of the western education system, and indeed in the case of Japan goes back much farther to at least the 6th century, with the introduction of Chinese models for society and the increase in Japanese going to China and returning with new knowledge to teach to their fellow countrymen. Education develops from there through the Heian era (794-1185) with the establishment of government institutions for study, aimed at the upper class. It is also likely that some people not in the upper class had ways of studying but at that point in time records are scarce as they relate to the lifestyle of commoners.
Skipping forward in time to the Edo Era, which is classified as the Early Modern Era and lasted from 1603 until 1867, the educational systems experienced a great growth for all people regardless of social status.
At the time, the government was organized so that the Tokugawa Family (from which Shoguns were chosen), had direct control over large parts of the country, but still most of the land was controlled by various Daimyo, prominent samurai families who passed their titles and control of their territories through their respective families. The Shogun had the power to remove Daimyo who failed to uphold standards of behavior and rule, and also retained the power to manage foreign affairs amongst other things, but the Daimyo were left otherwise to govern their domains in accordance with their own principles. The result of this is that there was not one unified government sponsored education system, nor a unified private system throughout the country. However, individual domains generally followed the lead of the Tokugawa Shogunate and so their systems in practice are similar.
The first type of school was the Hankou, official schools ran by individual domains and aimed at members of the samurai class. These were run very similarly to the Shogunate schools ran by the Shogunate, so for the purposes of this response you can consider them as essentially identical. These included various levels of education, of which some were similar to western universities in that they accepted adults who were looking for a higher education to advance their career or were otherwise interested in studies for personal reasons. They all had a basis in confician ideology, and taught many subjects, including writing skills, literature, geography, math, economics, foreign language, technology, etc... The yushima seido in edo, which was the most prominent school run by the Shogunate, is a famous example which produced many famous scholars and academics.
The second type were Terakoya, which were aimed at non-samurai children and run privately, most often by the teacher. They had no specific curriculum, but generally taught reading, writing, arithmetic, and history, and were usually attended by children who would now be elementary school age. Boys and girls could attend together, and teachers could be commoners, samurai, doctors, monks, priests, and so on. Schools particularly for certain skills, like music or art also existed, and schools specializing in teaching useful household skills to girls existed as well.
The last major group is the Gougaku, government run schools of various kinds that were seperate from the main Hankou. Some provided education to samurai, some to commoners, and some to a mixed group of both. They are distinguished from Terakoya and private institutions because they were established with government support.
Beyond that, various private institutions and personal tutors also existed, but that has changed little to this day.
So, what changed?
The majority of these institutions disappeared when their government backing disappeared, or in the case of Terakoya, when universal education supported by the government came into existence.
The subjects and methods of teaching changed in many ways with the adoption of western ideas which challenged confucianism.
Admissions practices changed, with the new universities being open to anyone in the country regardless of status or place of residence.
The internal structure, with concepts like choosing a major and graduating with degrees such as a bachelor's degree, a graduate degree, a masters degree or equivalent was a new addition not present in the previous system.
The idea of paying a fixed fee for a set course was introduced
As a result, essentially every educational institution in Japan is based on the western model and very few remain from before then. Additionally, since the naming schemes are different, the first college in Japan would be the first place to be called a college, regardless of the existence of any equivalent institutions.
Therefore, even with a long history of education, the modern system derives mostly from the western system, and thus the institution of higher education in Japan that first based itself on a western model to some extent is the first college.
Incidentally, that would be Keio University, founded in the final years of the Edo Era.