But I couldn't find information on what makes Taiwan so unique. Wikipedia has a Semiconductor industry in Taiwan article that has some historical information, but falls short of providing what sets Taiwan apart.
How did this tiny little island end up leading this strategical market so profoundly and why cannot much bigger nations compete?
This is because semiconductors have economies of scale over extremely large number of units produced.
Economies of scale mean that the more you produce the cheaper production gets. Many firms will have economies of scale over some range of production, but it is very rare to have economies of scale over very large quantities. In such cases you will typically observe very heavily concentrated industry or natural monopolies. This is usually due to high fixed cost.
Semiconductors are one of the industries with economies of scale over very large numbers of units. This is because a modern semiconductor fab/foundry costs around 20 billion dollars (as reported by the Economist). A typical chip manufacturer will have several foundries not just one. Hence, chip making is not something you can simply do in your garage like, let's say, building computers. You need to produce an extremely large volume of chips just to break even and the more you produce the more profitable you get. Of course, at some point there would be diminishing returns but when it comes to semiconductors that level of production is hard to reach.
As to why it is Taiwan and not some other place there are several reasons. Some important ones:
Taiwan had a relatively highly educated, but at the same time cheap workforce in the past when the industry started.
Taiwan had good institutions (e.g. rule of law). Having good institutions is a general precondition for development (e.g. see Acemoglu and Robinson 2012).
Taiwan managed to strike a good balance between providing support to tech industries via grants and building infrastructure yet did not protect the industry from competition or try to nationalize it or over-regulate it or micromanage it too much (see Tung 2002).
Open economy: trade and foreign investment was promoted and encouraged.
Luck. There were multiple places where this could have occurred as well. Life is not completely deterministic; random chance plays a great role. If we were to reset the clock things might turn out differently even keeping all the policies the same as before.
The semiconductor industry is unusual in that it is driven by a few individuals who innovate the technology. If you read up on the early history of the industry and William Shockley, you get an understanding of how much influence a few individuals had. Many people believe that Silicon Valley won over as the tech center over many competing locations because Shockley and a small group of his engineers decided to locate there. To understand why a particular location dominates in a field, it may be useful to understand why Silicon Valley won over places like Boston and the Research Triangle, places that invested heavily in computer technology but lost out to SV.
To understand why Taiwan has the lead in semiconductors today, you have to read up on Morris Chang. I'm going to give a controversial answer to your question but even if one does not agree with it, you can't hope to get any clarity without reading up on MC. His wikipedia page doesn't really give insight so I would look at other sources.
Early in his career, Morris Chang worked at Texas Instruments. He was considered brilliant and almost solely drove leaps in the success of TI. Due to space, I won't get into it but you should read up separately on his achievements there. MC ended up becoming a VP at TI and was considered a shoo in for CEO. When the opportunity came up, the company unexpectedly chose another person for CEO. Chang quickly left TI after that and ended up forming TSMC in Taiwan. TSMC is largely the reason Taiwan is leading in semi production.
Presumably, he was denied the CEO position due to his race. It can't be proven but there isn't any other compelling reason. He wasn't born in Taiwan and had never been there prior to taking an offer to start TSMC there. So he didn't have any special affinity for the island. TSMC should have been a US company. MC studied in the US and had been there since the age of 18.
So the controversial answer to why Taiwan leads in semis is racism. Sergei Brin's family left Russia due to anti-semitism. He ended up founding Google. To believe that one individual could have such influence may be difficult to believe but with enough research and understanding of the field, it becomes the most plausible.