Strange question I know. But does Touhou have a solid overarching story? With all this talk about fan games and what seems to be the sheer amount of content in the series. I wonder how exactly that whole thing fits together.
Touhou has a continuity, but it's mostly episodic. Basically, each game tells the story of a specific "Incident;" That is, a deviation from the norm. The seriousness of the Incidents can vary, from something as harsh as a never-ending winter to "WOAH, RANDOM FLYING BOAT!" Generally, it's not overly story-heavy, with a bit of background info and in-game dialogue explaining a bit about the characters and their motivations. Still, there's not that much to go off of, giving a lot of room for fan interpretations to run wild.
The mangas are usually self-contained, but characters often cameo in them, making it clear that they're all in the same continuity. Generally, they kinda just drift in and out on a whim, though sometimes characters land major roles in the manga's plotlines. Said mangas also vary in terms of narrative, from the loosely-connected gag manga
Inaba of the Moon and Inaba of the Earth to the serialized
Foul Detective Satori (Which has unfortunately been postponed due to the author's health issues).
While the series doesn't make many major changes to the status quo beyond "Hey so these characters exist now," there have been some story arcs where one game leads to another. There's two instances of this:
The tenth game,
Mountain of Faith, introduced the Moriya Shrine and the gods Kanako and Suwako. In Touhou 11, Kanako gave a hell raven the power of
nuclear %$#@ing fusion, who created a geyser with their powers. Said geyser blasted the flying boat I mentioned earlier into the air, starting the plot of Touhou 12. The flying boat then landed above a mausoleum containing a Taoist hermit who is also the real life Prince Shoutoku, who then woke up for the plot of Touhou 13. If you want to stretch it beyond that, Reimu and the factions introduced in the 12th and 13th games played a major role in the spinoff fighting game
Hopeless Masquerade, with the Taoist hermit outright creating that game's antagonist.
Later, another spinoff fighting game,
Urban Legend in Limbo, introduced Sumireko Usami and the Occult Orbs, which casually
busted a hole in the barier around Gensokyo. While that's already a jaw-dropping feat, the 15th mainline game revealed it was actually a plot by the moon people to stage an invasion in Gensokyo to make the Lunar Capital manifest inside Gensokyo using some weird concept frickery I don't quite understand. The next fighting game,
Antimony of Common Flowers, followed up on the Occult Orb subplot, and also introduced the dream realm, which was heavily featured in a spinoff title that heavily featured Sumeriko.
So yeah. Bottom line, Touhou is episodic and leaves a lot of room for fanworks to run wild, but it also has a tendency to casually reference past events and characters whenever just because it can.
Also Touhou logic is really weird and if you don't understand it that's fine because I don't either