"Tokyo Metro is operated by Tokyo Metro Co., Ltd. (東京地下鉄株式会社 Tōkyō Chikatetsu Kabushiki-gaisha), a private company jointly owned by the Japanese government and the Tokyo metropolitan government.
The company replaced the Teito Rapid Transit Authority (帝都高速度交通営団 Teito Kōsokudo Kōtsū Eidan), commonly known as Eidan or TRTA, on April 1, 2004. TRTA was administered by the Ministry of Land, Infrastructure and Transport, and jointly funded by the national and metropolitan governments. It was formed in 1941, although its oldest lines date back to 1927 with the opening of the Tokyo Underground Railway the same year."
https://en.wikipedia.org/wiki/Tokyo_Metro
Most Japanese cities' subway systems are not private, and even many of the "private" train lines were built using extensive public subsidy.
Japan also runs a massive defecit relative to GDP (~250% vs our ~100% value). They spend like drunken sailors on public infrastructure.