Edit: I should have written 'East Asian' in the title or tried to be more precise. To be fair, I am not 100% sure this trope even exists in fiction of most other East Asian countries. I came across it in Japanese manga/anime and in Korean manwha. I feel like I came across it in Chinese works, too, but can't seem to recall any examples.
Oh, man. I love this question. So.....
There's actually a couple of different pieces that add into this long running trope.
First, when it comes to the history of martial arts, the classic division of martial arts schools and types harkens back to the old dichotomy of “Shaolin” and “Wudang”. This was initially a divider of different types of styles of Chinese martial arts (you can read more about it here in this thread ). However, by the late 1800’s, there was a plethora of different styles of martial arts across China, but particularly in the southern areas of Guangdong and Fujian. There are a number of reasons for the mass proliferation of martial arts styles and teachers in the southern provinces during the 19th Century. It was a very chaotic time with violent incidents like the Opium Wars, The Taiping Rebellion and the (concurrent with the Taiping Rebellion) Punti-Hakka Clan Wars. This led to a rise in the teaching of martial practices both for defense and profit. As more and more teachers moved into the cities, there was competition for students. A public fight or competition could drum up business.
Second, many of the martial arts “schools,” including many that are still practiced across the world today, were “family” styles. That is to say, they were crafted by an individual(s) and the style took their name. They were often disseminated through the creators family before they reached other practitioners who would then go on to teach the style as well. So, if someone were to question the ability of your family’s style, you might be inclined to respond in a very forward way.
Third, the image that most of us have in our minds when we think about these school rivalries in Hong Kong fantasy action films or Japanese fantasy manga/anime is very much influenced by styles of fiction that came before it. In the Chinese realm, it was very heavily influenced by the wuxia novels of Jin Yong (a.k.a. Louis Cha). His fiction works, which started in the mid 1950’s, really played on the idea of powerful, semi-secretive sects of martial artists that were in competition with each other in a consistently running good versus evil kind of match.
Finally, and this comes from just personal experience and knowledge, martial artists are very similar to sports fans. They get very, very attached to their school (team) and will get very incensed at any disparaging of their school (team) and love to show off when their school (team) is great. However, with a sports team, you don’t really have much control over how they perform or what they do. As a martial artist, your ability very much rests with you so you can attempt to assert your claims right then and there.
Further Reading:
Hamm, Christopher. Paper Swordsmen: Jin Yong and the Modern Chinese Martial Arts Novel. Honolulu: University of Hawaii Press, 2004.
Henning, Stanley E. “The Martial Arts in Chinese Physical Culture, 1865-1965.” In Martial Arts in the Modern World, edited by Thomas A. Green and Joseph R. Svinth, 14-33. Santa Barbara: Praeger, 2003.
Judkins, Benjamin and Jon Nielsen. The Creation of Wing Chun: A Social History of the Chinese Martial Arts. Albany: State University of New York Press, 2015.