In one such area on the edge of a vast bog, in the shadow of some jagged hills, lived an elderly couple: Axl and Beatrice. Perhaps these were not their exact or full names, but for ease, this is how we will refer to them.Now, do you react to this detail by thinking "their true names and identities will be deeply important before the book is over, and I should start ruminating on Arthurian characters" or do you think "huh, names, OK" and move on?
(Oh, one other point: Axl always calls Beatrice "princess." In nearly every line of dialogue. He never calls her Beatrice. Could that be important?)
I'm sorry to inform you that the latter is the correct answer; that is a Chekhov Gun that Ishiguro very carefully places on the mantelpiece at the beginning of the book and then entirely ignores from that point forward. Axl turns out to have been a mid-level soldier in Arthur's armies, actual role and title left unspecified though he did do some peace negotiations, and his full name is Axl-somethingorother. And Beatrice is just his wife.
That's a key issue with The Buried Giant: it's written in a slightly too-fussy style, it doesn't manage its genre elements particularly well, and it's clumsy in general. It's not maddeningly wrong-headed like Ishiguro's attempt at science fiction, Never Let Me Go, but it shows an essential lack of knowledge about how to present fantasy and an instinct to dull everything down and make it more tedious.
Ishiguro has a powerful metaphor at the center of The Buried Giant, and keeps obscuring that metaphor, or letting it make his book duller and less focused than it should be. Some kind of mist -- we find out the source before the end of the book -- has been making the people of this area forget their pasts. Axl and Beatrice think of this as a curse, and want to eliminate it, if they knew how. But they're old and mostly forgotten themselves, living in a minor subsistence-level village in the middle of nowhere, and they're more concerned with the fact that the rulers of their village no longer allow them to have a candle in their room at night.
But, after much hemming and hawing, and much Alphonse-and-Gaston dialogue between our old couple -- "do you want to do X, princess?" "only if you do, darling Axl" "well, I only care for your happiness, princess, so my concern is purely for your needs" and on and on and on -- they actually leave their nameless village and head across the nameless lands nearby to try to find their nameless son in the nameless village he moved to some unknown number of years ago, after he left home for some unremembered reason.
All of that namelessness and forgetting is due to the mist, of course. But Ishiguro lets it be a drag on the narrative and on his characters: it keeps him for describing the world crisply and keeps his story muddy and rambling. His tone doesn't fit a rambling, loose story, though, leaving the sense and matter of The Buried Giant continuously mismatched.
Much of what was forgotten was a brutal war between Arthur's Britons and the (invading?) Saxons, and one particular war atrocity that we learn about at second-hand, in a dull, muted form, late in this book. (If anything in The Buried Giant called out for a flashback, it was this: but it doesn't get one, or any other mechanism to make it live and command attention.) Nowadays, Saxons and Britons live amongst each other, both in scattered separate villages and even together in some of those villages. Remembering the war could threaten that peace -- but by the time Axl and Beatrice learn that, it's too late.
That, probably, is Ishiguro's point: relying on forgetting about things to make them better only works for as long as the forgetting does. And remembering can spread quickly. But, then again, the loss of the mist will probably lead to mass bloodshed -- the return of the wars at least, and possibly just straight massacres.
In their travels, Axl and Beatrice meet a few others whose paths cross and join theirs for a while -- the old Arthurian knight Gawain (who has no particular reason to be Gawain, and is just "an old knight who was sworn to King Arthur"), a Saxon warrior on a secret mission, and the Saxon boy that warrior is forced to foster when he loses his parents to monsters. There's also a dragon, Querig, supposedly terrorizing the surrounding area, though she hasn't been seen for years -- and Gawain's mission for the past however-many years is to kill Querig, which he claims he's on the verge of doing any time now.
As usual with Arthurian stories, there's an attempt to drag Christian morality into it, with a community of monks and their dark secrets. But, again, its connection to the rest of the book is muddled and confused; it's just one more damn thing Axl and Beatrice have to get through on their journey. (In a weird way, The Buried Giant is a frustrated and depressing picaresque novel -- they go to strange places, meet odd people, and keep learning more about how horrible everything is.)
There's a very good novel lurking inside The Buried Giant, and that seems to be what most of the mainstream critics reacted to when they read it. But it's obscured and damaged by the choices Ishiguro made, by the ways he deals with his material, and by his ignoring or flouting useful fantasy expectations and revelations. This is a fantasy novel for non-fantasy readers, for the simple reason that non-fantasy readers will appreciate what The Buried Giant does well, and not realize all of the things that it misses doing or trips itself up on.