Is Shanghai Stock Exchange a listed company?

No, it's a national unit that manages and supervises listed companies!

Shanghai Stock Exchange, the full name of which is Shanghai Stock Exchange, is a non-profit-making legal person that provides places and facilities for centralized securities trading, organizes and supervises securities trading and implements self-discipline management. The Shanghai Stock Exchange is directly managed by the China Securities Regulatory Commission, and together with the Shenzhen Stock Exchange, it is called the two major stock exchanges in Chinese mainland. It is also a member of the International Organization of Securities Commissions, the Federation of Asian and Oceania Exchanges and the World Federation of Exchanges.