In The Hobbit and Lord of the Rings books, Tolkien manages to cultivate a gravity around many of the fictional items and creatures he's invented. In retrospect, it starts with the titular ring itself, The One Ring as it's notated in the RPG and wargame derivatives, but it includes famous creatures like Smaug and Sméagol and Sauron, and items like Orcrist and Glamdring. In Tolkien books, there is very often exactly one, or three, or nine of something, and no more. Middle Earth isn't a world where people raise and train dragons or use magic to tidy up after tea, it's a mostly mundane world, if you ask the Hobbits of the Shire or the menfolk of Dale or Rohan. So when you encounter a mythical beast or legendary item in Middle Earth, it's a significant life event, and not something you're likely to ever forget.
In The Hobbit, Tolkien establishes the model for Middle Earth legends. There are things in Middle Earth that are wholly unique, or at least functionally unique. Smaug isn't just a dragon, he's the dragon. For all anybody knows, Smaug is literally the only dragon in Middle Earth (we learn later, in Lord of the Rings, that there are, or have been, other dragons, and Ancalagon the Black is named specifically by Gandalf).
The ubiquity of Smaug helps define the regions of Middle Earth. Whether accurate or not, Smaug is the dragon to the people of Eriador. They didn't think about "dragons, including the one named Smaug". To them, the word "dragon" meant Smaug, the way the word "pandemic" for us in 2024 is synonymous with "COVID 19".
Knowing that Tolkien was a scholar who translated the epic poem Beowulf makes me think that Smaug is a little like Grendel. It's way over my head, but as far as I know, we don't really know what kind of monster Grendel was meant to be. The Internet tells me he was probably a humanoid giant, but it seems the original poem leaves it open to interpretation, or rather to expectation. The reader was expected to know what kind of monster Beowulf would have had to defeat. It would be the kind of monster you've heard about already, the one in the collective consciousness of your community. When people said "monster" in the fictional world of Beowulf, it was synonymous with Grendel (or at least, Grendel and his immediate family, most notably his mother), just like when people in our real world past read about a monster named Grendel, they apparently knew what to imagine without a detailed description.
In other words, sometimes (or maybe most of the time) the uniqueness of a monster in Tolkien's world doesn't say much about the monster, it's about the people living in fear of the monster.
It's not just creatures that get special names in Middle Earth. Swords and heirlooms and magic items do, too. Glamdring and Orcrist and Sting and Andúril are profoundly significant to any reader because they're iconic. We know them by name.
They're equally important to the fictional world of Middle Earth, because in Middle Earth those are singular items. Andúril is a symbol of royalty, but also of chivalry. You don't just get Andúril because your dad owned it. You get Andúril because you've earned it (and also because your dad owns it). If you wield Andúril, that says something about you as an individual. (Hey, it's fantasy, so the myth of innate nobility and bravery can be true in Middle Earth.)
And there's only one Andúril. You can imagine the honour it would be to see Andúril carried by Aragorn. It would certainly be at least like meeting your favourite celebrity today, or like seeing a rare artefact at a museum. It's the only one in the world, and you've seen it. That might not make you special in any sense, but it does make you a step rarer. Not everybody has seen it, but of those who have, you're one of them.
It also makes the items all the more precious (pardon the choice of wording). If you lose Orcrist, it's gone. There's no more Orcrist. If you destroy The One Ring, it's gone, there's no more One Ring because it was the only one of its kind.
Tabletop roleplaying games mimic this, sometimes, by listing very specific magic items in Game Master guides or in source books. The same goes for creatures. There's one Cape of the Mountebank, so when you find it you have to get it. That's likely your one chance in that campaign. It's a big deal, it'll cause you actual stress.
The singleness of an item is important, and using it wisely in games and fiction is a powerful, and delightfully manipulative, creative tool.
In Two Towers, Sam has the dubious honour of seeing the legendary war beast the Hobbits know as an oliphaunt. This is Tolkien, though, so Sam doesn't just see any oliphaunt, he sees Mymak, a fierce oliphaunt of notoriety among the Gondorians.
Why does this oliphaunt deserve to be reffed to by name? We don't get Mymak's backstory. We never find out what happens to Mymak. Mymak's mother doesn't show up for retribution. But we know that the oliphaunt observed by Sam and Frodo and Faramir (and company) is named Mymak. Why?
I think the obvious and simple answer is "flavour". By giving a name to a random oliphaunt, Tolkien relays to us that there are lots of oliphaunts, some more terrible than others, but this one, the one in this chapter, it's so bad that people recognise it on sight and have a name for it. Don't be fooled by Isengard's fall, the fact that Mymak has joined the fight is bad news for Gondor and everybody else.
But I think Mymak means more than just bad tidings.
Sam had only heard of oliphaunts in a schoolyard rhyme. He wasn't sure whether oliphaunts even existed. Seeing one is a big deal, but it's just one of many. Mymak isn't the oliphaunt, Mymak is just an oliphaunt. By giving it a name, even though the name actually means nothing to us, the oliphaunt we see along with Sam becomes more than just another war beast. We don't know who Mymak is, but we know Mymak has a reputation, and that gives us a peek into a larger world, with a life and history all its own. That we don't know its whole life and history makes no difference (in fact, if anything it arguably makes it all the more intriguing), we know that Mymak exists and has existed in it.
Tolkien somewhat uses the same trick when we meet the Witch-King of Angmar. Breaking all the rules of plot development, Tolkien seemingly invents a major villain for us just moments before he appears. We didn't really know or care about the Witch-King until Return of the King, but his confrontation with Éowyn is one of the most memorable (although no better written, in my opinion) battles of the entire book series. There are lots of Nazgûl and then there are Nazgûl, and we understand that this one is important because he's got a "name" (such as it is) and a backstory and a reputation. Essentially, we care because Tolkien broadcasts that it's a big deal.
I don't think this trick works universally. It's a Tolkien thing because, I think, he has the lore to back it up. He's a talented bluff, Tolkien is. He's got a hand of cards and manages to convince us that they're cards we've never even heard of, but that they're very very good. When you try and call his bluff, he actually does have the card he claims to have. He's got pages and pages of notes and not just a few letters from 1947 about it. It was a thing he's been meaning to tell you about for years, but just never got around to writing a whole book around it.
Sometimes, though, you call his bluff and he admits he was bluffing all along. Mymak is just a name he made up to make an oliphaunt sound important.
So you learn not to call Tolkien's bluff. If you call his bluff and you're right, you spot a rare crack in the world of Middle Earth. But as long as you believe Tolkien's mastery of his own lore, you can continue to be lost in Middle Earth's never-ending depths.