How do we know what the shiny variations of legendary pokemon look like if there is supposed to only be one of each species? And how do we know their typings if it’s so hard to get close to one? Is the “one of each” thing simply a myth telling us how rare these pokemon are?

Legendary pokémon are rather unique in that there are actually an abundance of depictions of them via ancient art. Remember, legendaries are essentially gods or powerful spirits to many cultures, and as such, many humans have created paintings, idols, statues, and so forth depicting what legendaries are normally like. We can gather that the images we see are of “normal” variations if that legendary is of a particular color in many of its depictions. For example, we assume that Ho-oh’s normal coloring includes blue, green, and violet because most of the paintings in Johto depict it as such. Thus, if we ever see Ho-oh without those colors, we can safely assume that this is a variant color, not its normal coloration. (Either that, or our ancestors were wrong.)

As for typings, we don’t always know for certain, although we can collect data from exceptionally strong trainers who have managed to encounter or capture one themselves. For those legendaries we still have yet to catch (Arceus and Ho-oh, for example), the officially listed typing is more or less a hypothesis based on what we know from their respective legendaries. Ho-oh, for one, is listed as a fire- and flying-type because it’s a bird known for its sacred flames. Lugia, by contrast, is a flying- and psychic-type (as opposed to water or dragon) because the legends about it describe abilities that are more in line with psychic-types than water- or dragon-types … and also because we have data from trainers who have managed to get close enough to Lugia to observe it.

Finally, as for your question concerning uniqueness … that’s a difficult one to answer. For some legendaries, it’s very true that there are more than one, and the uniqueness the pokédex emphasizes is really just a label, as you say. This is the case with Latios, Latias, and Lugia (although the lugia Johtonians worship is a very specific one, not just any lugia—and lugia are extremely rare either way). Meanwhile, for other, more powerful legendaries (such as Arceus, Dialga, Palkia, and so forth), it’s very likely there’s only one. Then there are legendaries who are somewhere in between—that is, we don’t have enough data to determine whether or not they’re unique. This is the case with legendaries such as Mew or even Ho-oh.

In short, much of what we know about legendaries is tied to conjecture based on legends and folklore. In reality, we don’t have a lot of data to know for certain if many of our assumptions—of populations, of powers, and even of color—are completely true.