This house looks amazing! But it doesn't seem like a "tiny house". I thought tiny houses were by definition on wheels with a more flexible way to hook up and out of utilities?

I don't think so. Rather, typical zoning codes in the US don't really allow for permanent tiny homes, so they usually have to be built on trailers in order to be legal.

I've seen plenty of "tiny home" projects built out of old shipping containers for example.

Not necessarily. Many are stationary, built on standard foundations. Though given their size, they are definitely easier to move than larger buildings.