Fate/Stay Night is a visual novel about ideals and different paths on which a young man chooses to pursue the life of a hero, whatever he determines that word to truly mean, and for that reason, there is no canon route. But precisely because of the Nasuverse's inclination toward alternate universes and timelines, it wouldn't be impossible to tell a story about Shirou Emiya's life after the events of the VN.
In fact, the epilogue of Unlimited Blade Works already paints a compelling picture of what a full Fate/Stay Night sequel might look like, where Shirou and Rin go to study at the Clock Tower of London to hone their magecraft. So why hasn't there been a sequel about Shirou's life after the Grail War, and better yet, has Shirou appeared in anything else?
Where in the World Is Shirou Emiya
To start, let's disregard stories that are clearly separate realities wherein the events of Fate/Stay Night never happened or were heavily altered, such as Fate/Kaleid. Looking only at the stories that remain, Shirou doesn't show up much at all. He's mostly relegated to cameos in stories like the recent novel series The Adventures of Lord El Melloi II.
Just like in the epilogue of UBW, these cameos suppose that Shirou went to London with Rin, studying and working part-time as a butler for Luviagelita Edelfelt, much to Rin's chagrin. But these cameos are just that and don't really give much of his perspective as he's in another's character's story. But there's so much potential in a college story centered around Shirou and Rin.
Episode 25 of UBW was so different from the rest of the series that it may as well have been a backdoor pilot for a new series. Perhaps if not for the Heaven's Feel films, it might have gotten made. After all, most of the episode was original content that wasn't in the VN, and as far as creative liberties go, it made for quite a story pitch.
As an end to the series, it ties up loose ends comfortably, but also introduces so many new things along the way. The tone was light and fun and the way Rin's rivalry with Luvia devolved into a magical MMA duel was hilarious and badass in unexpected ways. To top it all off, Shirou meeting Lord El Melloi II, a grown-up Waver Velvet from Fate/Zero, was a hugely satisfying and symbolic union between the survivors of two Grail Wars.
Imagine a full series centered around Rin and Shirou's studies, learning under professors like Lord El Melloi and perhaps crossing paths with characters from all corners of the Nasuverse. If it's anything like the epilogue, it wouldn't be too dissimilar to Lord El Melloi II's Case Files; a fun adventure series with some comedic charm, fascinating world-building, and occasional character drama.
It's a chance to see Shirou's path to realizing his ideals, but with the support of Rin to make sure he doesn't make the same mistakes that Archer did. On paper, it sounds fantastic, and what has been shown of this concept is nothing but impressive, but there are also reasons why it's likely an idea better conceived than executed.
Why Hasn't It Been Done?
UBW's epilogue was perfect for that route, but part of the charm of Fate/Stay Night is how there is no definitive path through the story. Each route sees Shirou either reaffirm or challenge his beliefs in ways that expand on an already solid foundation built on his trauma, ideals, and aspirations. To tell a story set after the Grail War almost seems like you're espousing that there is a true ending.
With that said, it's not impossible to create a Fate/Stay Night sequel that respects the different routes and their lessons. After all, they already kinda did it, albeit in a way that an outsider might view as cumbersome. A past piece about the "canon ending" covered Fate/Hollow Ataraxia, a sequel story that aims to combine all three routes, leading to a timeline where many of the characters live past the end of the war.
Remember The Adventures of Lord El Melloi II as mentioned before? That story isn't technically a continuation of UBW's epilogue, but a hypothetical fourth route of Fate/Stay Night where all three routes happened. It's a lot to take in for certain, but it's definitely one of the more complete ways to set up a continuation.
Arguably, it puts Shirou in a perfect position to start a story, with the memories of three different outcomes to the Holy Grail War and three different paths he could have taken. Now he's free to decide what lesson to take from the whole ordeal while he learns plenty more, and alongside not just Rin, but the whole surviving cast of Fate/Stay Night.
I suppose the message at the end of all of this is that there really should be an adaptation of Fate/Hollow Ataraxia. Secondly, even with the potential risk of sullying Fate/Stay Night, it would be truly delightful to follow Shirou on a new adventure that hasn't been written before, to take advantage of this universe's infinite possibilities. For better or for worse, it could be a great time.