The ‘why’ question is indeed the interesting one, not least because it isn’t clear what the actual answer is.
As WhyNot points out, ermine was expensive. It was also exotic. Stoats’ fur only turns entirely white in the far north, so in most parts of Europe ermine had to be imported. But it was just one of many types of expensive, exotic, imported furs known in medieval Europe. Moreover, fur-lined clothes were luxury items with an obvious practical advantage. Being able to wear fur, particularly of high quality and in bulk, was a perennial status symbol.
Yet in the fourteenth century ermine-lined robes seem to have emerged as a stereotypical royal garment. By the 1390s, for example, Richard II was dressed to look like a king. Or at least dressed to match our idea of what a king should look like.
Part of that was doubtless because there was much more fur about. Imports of fur from northern Europe increased rapidly during the fourteenth century and there were particular fashions in fur-lined clothes. Kings tended to wear fur because that’s what other rich people were wearing. But that doesn’t get us very far. It can hardly be that ermine becomes specifically royal because lots of other people were wearing furs.
Nor do sumptuary laws provide much of an answer. Those usually did not restrict ermine to royalty. Thus the 1363 English statute specifically allowed wealthier knights and above to wear it.* It was certainly ‘high status’, but not exactly royal. The fur that the 1532 statute restricted to the Royal Family (and, under certain circumstances, to dukes and marquesses) was not ermine but sable.
*Not that one should make the mistake of assuming that English law can stand for what applied elsewhere, but the general point does still seem to be true.
Moreover, it can be difficult to tell when kings wore ermine-lined cloaks just because such cloaks were reassuringly expensive and when they did so because ermine had become literally a status symbol. Eventually, of course, they would become an example of a form of dress that could only be symbolic because it was so obviously old-fashioned. But there may also have been an element of kings (Charles V of France?) self-consciously adopting a style they could make regal.
Perhaps the real answer is that ermine was so easily identifiable. Those black tails (already more often black lamb’s wool) were distinctive and, most importantly, showed up well in painted representations of such robes. I do rather suspect that it’s not just that kings wore ermine and that therefore they continued to wear it, as that ermine becomes part of the standard iconography of what everyone thought kings should wear, thereby encouraging kings to live up to that expectation. So long as there is the idea that kings should look like kings, they will appropriate - or have appropriated for them - any convenient symbols, whether crowns, sceptres, military uniforms, horse-drawn carriages or corgis.