The text says Sai Baba has 'no name or village.' Does this mean his identity was completely unknown and he was simply an anonymous wanderer?

While Chapter 36 of the Shri Sai Satcharitra states that Sai is one 'who has no name or village,' this is intended to describe his transcendent and detached spiritual nature, not a literal lack of identity. The passage highlights his spiritual state, being beyond worldly attachments. The text clarifies this by calling him the 'incarnation of the knowledge of Truth' who stays 'distant and detached from his name.' He was, of course, known as Sai, and Chapter 36 confirms his 'infinite glory' was such that he could 'turn a beggar into a king in a moment with the play of his eyebrows.'