You may not get the best answer here since the issue has probably been studied more by sociologists than historians and it bumps up against the 20-year rule. I think part of your question is "Why did StarCraft continue to have a substantial following in South Korea well after most players in the rest of the world had moved on?" which would require discussion of what happened more recently than 2002. The sources I have consulted for this are journalistic accounts and sociology articles and papers.

So I can tell you a little about how StarCraft became so entrenched in South Korea shortly after its March 1998 release. Basically it comes down to timing. StarCraft came out at a time when South Korea was uniquely fertile territory for a competitive online game to become very significant. This is in part due to the material context in which games were played at this time.

First a little bit about StarCraft. I assume the OP knows this, but for all readers' benefit, StarCraft is a real-time strategy (RTS) game published by Blizzard Entertainment. The gameplay resembles that of Blizzard prior RTS games WarCraft and WarCraft II. StarCraft provided a new and original scifi setting (though with many echoes of familiar properties) featuring three substantially asymmetric factions. Each faction was quite different to play, but they were generally regarded as balanced against each other. Although the game has a pretty well-regarded single player campaign (with cinematic cutscenes and memorable characters), it also has easy-to-use LAN and online (internet) multiplayer modes, the latter fully integrated with Blizzard's Battle.net service at no additional cost (if you owned the game itself and had an internet connection). RTS was a pretty hot genre at this time and online competitive multiplayer was an established way to play; however, Battle.net's integration directly into the game was new and, at the time, unique to Blizzard's products. With Battle.net players would quickly and seamlessly be put into an online game, in contrast to earlier systems that required a separate client and layers of preplanning. E.g., a competing game, Command & Conquer from Westwood Studios, had similarly challenging gameplay but online play was a chore to set up. So Blizzard had created an RTS game that provided a very good online multiplayer experience. That is pretty much all I'm going to say about StarCraft itself, except that Brood War was an expansion that came out in December 1998 and added some new units (and extended the single-player storyline) but did not fundamentally alter gameplay. Well, one more thing: in my opinion as a videogame addict, StarCraft (with or without Brood War) was a really good game and in competitive play has a real skill curve.

Blizzard had not particularly targeted the South Korean market and had not even bothered localizing the game for Korean-speakers, expecting perhaps only a few thousand sales. Instead, sales in South Korea during 1998 reached 120,000 and then over 1,000,000 in 1999. How did this happen?

To understand this phenomenon, we need to talk about the infrastructure of videogames in South Korea. In the early 1990s the South Korean government had decided to make a big investment in internet connectivity which included making broadband hookups available to many communities. This was helped by the fact a substantial share of the national population was in the Seoul urban agglomeration. By 1997, South Korea was heavily "wired," but home connections were still costly and had poor penetration. To provide access to ordinary people and increase interest in the internet, the South Korean government funded community access centers with computers connected to the internet--basically like free cybercafes.

Then the East Asian financial crisis of 1997 struck, throwing many South Koreans out of work, making savings disappear, and decreasing tax receipts. The government slashed funding for the community access centers. Into the void stepped entrepreneurs and investors who opened fee-based cybercafes called PC-bang.

To understand PC-bang, we need to talk about South Korean youth entertainment culture. Here I am talking the interests of South Koreans about ages from tweens to twenties, middle school to university/early employment, and also chiefly male. Comic books, movies, and videogames were popular with this demographic as in many other countries, but these interests were often pursued in businesses called bang, which is a Korean word meaning "room." Say you wanted to watch a scifi movie on DVD with your buddies from school. Instead of meeting at somebody's home and watching it there (hoping to take over the living room for a few hours and dealing with embarrassing parents, little siblings, "Have you done your homework?" etc.) you'd go to the local DVD-bang and view it there. If you wanted snacks, they were on sale. There were comic-bang, tv-bang, etc. The best place to locate a bang was, perhaps obviously, near a school campus of some sort.

So the PC-bang business model was basically to rent a big room near a school, fill it with decent gaming PCs, network them, acquire a small library of games, and wait for the money to roll in. Customers paid by the hour to use the PCs and of course the refreshment stand was open. Staffing costs were pretty low since owners figured (correctly) that the customers would know how to use and even perform basic troubleshooting on the computers and would take care of them.

Nationwide, there were 100 PC-bang in 1997. There were 13,000 in 1999. Part of this meteoric rise is a result of StarCraft, but part of what made the business model so attractive was the same financial crisis that had killed off the community access centers. A lot of people were out of work and looking for a hustle. A lot of people had been burned by the financial crisis (which had a bank failure aspect) and wanted to put their savings somewhere they could see. The sure thing they wanted was the PC-bang. (The financial crisis also probably increased the number of customers since there were more young people with time on their hands.)

StarCraft quickly became the dominant game in the PC-bang scene, despite not being in Korean (it was played in English). Its penetration was helped out by the distributor. Blizzard's titles had been distributed in South Korea by an arm of LG, but (again as a result of the financial crisis), LG sold the distribution to a startup headed by a laid-off LG executive. He went around the country promoting the game, giving free copies to PC-bang, providing posters, etc. StarCraft was a really good game for the PC-bang to feature since it had deep gameplay and, like I said before, Battle.net made any kind of networked play a snap.

Because this StarCraft-heavy videogame culture developed around the PC-bang, rather than home-based equipment, it overlapped with other social circles such as school and neighborhood. You would go to the local PC-bang after school and play against your friends, rather than go home and play against random people you knew only by usernames. Also, you might spend time watching your friends play (because perhaps there were no more PCs available at the PC-bang or you were out of money). Kids from the same school/neighborhood would know who had the good StarCraft skills. They would challenge each other to matches, according to some sources in lieu of fighting. Interestingly, concerns over PC-bang were incorporated into the sales pitches of telecommunications companies: basically, if you don't want your son spending time and money at the PC-bang, get broadband at home!

PC-bang owners saw this going on and sponsored tournaments both within the PC-bang and against others. This led to lucrative prizes, sponsored teams and, basically, the birth of modern esports, which the South Korean government was regulating as early as 2000. The concept of being so good at playing videogames that other people will watch you do it in a way that pays for . . . something . . . was proven in PC-bang and then scaled up.

So anyway that is how StarCraft took off in South Korea: it was really well-suited for symbiosis with the PC-bang business model and became a very significant part of youth culture.

I said I would not get into the longevity of StarCraft because of the 20-year rule, but I think you can see the outline here: StarCraft became entrenched, and not just as an emphemeral "hot new game" but as a cornerstone of PC-bang culture which overlapped with school and neighborhood.