I thought the rails were owned by the government, leased to the companies?

Some of them are, like the Shinkansen lines. Others are both owned and operated by private companies: https://en.wikipedia.org/wiki/T%C5%8Dkaid%C5%8D_Main_Line.

Additionally, the stations are generally owned by private companies—including the the development rights at the station. This means that the Japanese private rail companies capture a portion of the value created by the rail service, which otherwise would be an externality. So the companies have an incentive, as landowners, for rail ridership to stay high.