Shinto, Buddhism and Taoism have had a very profound influence on Japanese culture. These three religions and their influence on Japanese culture are also very much present in Gensokyo. While the indirect influence of these religions can be felt in many Japanese games, Touhou incorporates their presence in a surprisingly straightforward and whole way. It's not just singular references or things that kind of look or feel like things from religions.
Incorporating very real religions and especially their esoteric, magical side to a fantastic setting is something quite unique from a Western point of view. It would be very difficult for us to imagine a video game that wasn't some sort of historical strategy game that would feature religious conflict between Christianity, Islam and Judaism as a plot point. It would be even stranger to have a lighthearted game be about a conflict between Rosicrucians, Sufis and Kabbalists.
This shows how the Japanese have very different ideas about religion compared to the West. The country also has a tradition of gentle and humorous religious satire mostly about the rivalries of various Buddhist sects dating back to the medieval period. Japan also has a history of very real and bloody religious conflict and repression. As much as the religions present in Japan have inspired the culture, offered refuge, insight, beauty and deep appreciation into the surrounding world, they have also fostered corruption, strife, cults and been coaxed to serving imperialistic aims. All of these things come together in Touhou to create the unique depiction of these religions found in Gensokyo.
Touhou doesn't proselytize any religion, the print works often have quite a negative depiction of the practices of organized religion and ZUN's personal beliefs are largely unknown. The vivid, fascinating and magical depiction of these religions can however spark interest in them quite easily. If one starts digging deeper, they will find at the very least their understanding of Japanese and broader East Asian culture much deepend. For some, it could be the start of a lifelong transformative journey.
All of these religions and their depictions in Touhou deserve a much longer and detailed write-up, but we have to start somewhere. So let us start with a brief overview of Shinto.