The two places write the word with different characters. Kyoto writes it one way and Nikko another, and the difference is not a spelling quirk. They are not making the same thing.

A skin forms on the surface of heated soy milk. In Kyoto a skewer goes in at the edge and lifts it away as a single sheet, thin and slightly translucent. In Nikko the skewer goes into the middle of the skin and lifts it folded over on itself, so that what comes off the vat is doubled and correspondingly thick.

That thickness changes what the ingredient can do. The Kyoto sheet is a supporting player, added to a course for its texture and its pallor. The Nikko one is simmered in seasoned stock or served raw in slices like sashimi, and it holds the centre of the plate. The same soy milk, and two entirely different jobs.

The technique came from China with Zen monks, reached Kyoto, and was then carried around the country by mountain ascetics and travelling priests, which is how it turned up here at all. Nikko was a centre of mountain worship and of ascetic training, and the men doing that training did not eat fish. They needed something nourishing and easy on the stomach, in quantity. That is the likeliest reason the thicker version took hold here. Where you put the skewer turned out to decide what the food would be for.