Yes, Kei or "K" cars dominate the market here. This is because they are light and small so they don't damage the road surfaces nearly as much as normal cars and because they have lower emissions. Hence, road tax is much lower and the cost of insurance/shaken is too. The NBox has been a huge and growing success for Honda because although the engine is so small it is apparently pretty fast. Somehow, a bit like the Jazz really, it looks and feels a bit bigger than it is.
The majority are boxy housewife cars for city use but there are also jeeps, trucks, vans, mini bus style and sports car variants. They weren't allowed on motorways in the past but as they have got faster and better they are now allowed. I wouldn't like to do long distance in them but some, but not many, people do.
Our second car, when we had one, was a Suzuki Wagon R which was the leading Kei car for many years until the NBox took over. I didn't like it at all as it was very gutless and didn't even return good economy compared to my GD1 which although only one step up was at least a proper car and could do far more. Easy to drive and park though of course and that is part of their appeal.
I think the reason they have never taken off anywhere else is that no other government has given the format the tax and insurance breaks that they have here. Probably a mistake because I think they probably did their sums properly and they do cause almost no damage to the roads. The roads here are certainly in far better shape than the UK that's for sure.