What made it so unique and why was it a tragedy that it was destroyed?
It wasn't unique, and it was a tragedy only in the same sense that the loss of any library is a tragedy.
###The non-uniqueness of the Alexandrian library
For an American analogy, imagine what would happen if the Library of Congress burned down tomorrow. It’d be a disaster. Over 70 million manuscripts would be lost. But there'd still be the New York Public Library, the Boston Public Library, and all the university libraries. No books would be lost, because that isn’t how books get lost.
The ancient Mediterranean world had hundreds of libraries too. When a scholar in Rome wanted to do research on a topic, they didn't travel all the way to Egypt: there were libraries on the spot in Rome -- the Palatine library, the Atrium Libertatis, the Portico of Octavia, the Ulpian library, the library by the temple of Peace, the libraries at the baths of Trajan, the baths of Caracalla, the baths of Diocletian, Severus Alexander's library in the Pantheon, and more. There were plenty of libraries elsewhere too. Emperors and private citizens founded public libraries in all sorts of places: we've got evidence of large privately built libraries even in smaller centres, like Como in northern Italy, Timgad in Algeria, Prusa in Turkey.
Even in Egypt the Alexandrian library was of limited importance. For example, our sole surviving manuscript of one of Aristotle’s books, the Constitution of the Athenians, is a papyrus copy made at Heliopolis Hermopolis. It was made on recycled papyrus: previously it was used for farming records. That is, it was made on the cheap. No one was making a 400 km round trip an 800 km round trip to the library of Alexandria to copy it. Most ancient Greek books that have been found in Egypt come from Oxyrhynchus, even further away also a long way away. The book trade was thriving, there were libraries all over the place, and the loss of one particularly big library didn't suddenly change that.
Out of all the hundreds of libraries that existed in antiquity, not one has survived to the present day. Even if we imagine the royal archive in Alexandria wasn't burned in 48 BCE, the books there had no more chance of surviving to the present day than the ones in the Palatine library in Rome, which was well stocked, and had more consistent funding and state support over subsequent centuries. The Palatine library also had multiple fires -- it survived, though, for a while, thanks to governmental support: that is, governmental support is a stronger force than a library fire.
The survival of ancient books isn't something that depends on one repository: that would put them at the mercy of regime changes, shifts in governmental priorities, funding. Books survive if they were copied, repeatedly. The story of ancient books being lost isn't a story of library fires: it's a story of economics, long-term cultural developments, and above all, format shifts.
The books from the ancient Mediterranean that have today are the ones that survived the transition from uncial script to minuscule script in the 9th-10th centuries; those ones, in turn, are the ones that survived the transition from scroll to codex in the 2nd-4th centuries. Both of those transitions, and especially the transition from scroll to codex, caused much greater loss than the loss of one library could.
###The rise of a symbol
The question, then, is how the Alexandrian library got its iconic reputation, as a unique and irreplaceable repository.
That reputation is recent. Prior to 1980, the Alexandrian library was as often a symbol of vanity and excess -- the vanity of collecting tons of useless books that no one will ever read -- as it was of the loss of knowledge. Here's Seneca, in the 1st century CE (On tranquility of mind 9.5):
Forty thousand books burned at Alexandria: let someone else praise it as a beautiful monument of royal opulence, like Livy, who said it was a tribute to the elegance [elegantia] of kings and the nobility of curation [cura]. It wasn't elegance or curation: it was studious luxury -- no, not even studious, since they didn't collect books for study, but for a spectacle.
Similar opinions were dominant in the 16th-18th centuries, as part of a tradition of viewing learning as vanity: a 1979 article by Jon Thiem documents this tradition through Louis LeRoy's De la vicissitude ou variété des choses (1575), Thomas Browne's Vulgar errors (1646), Rousseau's Discourse on the sciences and arts (1750), and Gibbon's Decline and fall of the Roman empire (vol. 5, 1781, on the events of 391 CE; vol. 9, 1788, on the Rashidun conquest). Here's Rousseau, focusing on the (fictional) morality fable of Caliph Omar destroying the library in the 7th century:
if Gregory the Great had been in the place of Omar, and the Gospel in the place of the Alcoran, the library would still have been burnt, and it would have been perhaps the finest action of his life.
The present-day narrative of the library as a symbol of loss existed too, alongside the 'vanity of learning' tradition. But it only became dominant in 1980. The cause was Carl Sagan's TV series Cosmos. In eight minutes of TV Sagan invented out of thin air some myths that have come to be very widely believed:
He also repeated myths that he didn't invent: the idea that religion caused the onset of a 'Dark Age' and a millennium of superstitious ignorance; and the idea that there was still a library in the Serapeion when it was destroyed in 391 CE, which is doubtful.
Here's an Ngram that illustrates the impact of Cosmos. Sagan turned 'library of Alexandria' into a formal title, with a capital L. He also caused the rise of the phrasing 'Great Library of Alexandria', capital G and capital L, which wasn't a thing previously. You can see similar effects in other languages, if you plug in the appropriate phrases, though in most other languages the Cosmos effect fades after the year 2000.
###Further reading
Kiwi Hellenist, thank you so much for your scholarly work. A a career law librarian working mostly in Washington, DC I can tell you that just entering the Jefferson Bldg, one of many buildings at Library of Congress, I shed tears in happy wonder at the beauty of the place. PS. I spend a lot of time on Naxos Island in Greece. Again, thank you.