Why does the postmaster not have his own name in the story "The Postmaster" by Rabindranath Tagore?

To answer this question, we will have to understand Tagore's perspective on his beloved India. To him, India had long been mired in superstitious traditions, and he argued for a more open approach towards science and global influences. Tagore also deplored in his lifetime the constant conflicts between opposing factions of Muslims and Hindus; as an enlightened thinker, he argued for solidarity, understanding, and cooperation between religions and castes.
In one of his most famous poems, Tagore argued for independent judgment and freedom of thought. His message had always been the tearing down of "narrow domestic walls" that threatened peace, stability, and progress.

Where the mind is without fear and the head is held high Where knowledge is free Where the world has not been broken up into fragments By narrow domestic walls Where words come out from the depth of truth Where tireless striving stretches its arms towards perfection Where the clear stream of reason has not lost its way Into the dreary desert sand of dead habit (from Where the Mind Is Without Fear).

So, in "The Postmaster," the main character remains unnamed. It is entirely possible that Tagore meant the postmaster to represent a universal character, one who exists within the higher reaches of the caste system and who zealously protects the status quo at all costs. In the story, the postmaster abandons his faithful servant girl, despite her expressed desire to go with him:

When the postmaster had finished his supper, the girl suddenly asked him: "Dada, will you take me to your home?"
The postmaster laughed. "What an idea!" said he; but he did not think it necessary to explain to the girl wherein lay the absurdity.
That whole night, in her waking and in her dreams, the postmaster's laughing reply haunted her—"What an idea!"

The postmaster thinks Ratan's question "absurd." While Ratan sees her employer as a father figure, the postmaster feels no answering emotional attachment to her. The question of caste (and reputation) strongly permeates this story; thus, Tagore does not name his protagonist because he is highlighting a universal issue in Indian culture, one that continues to resonate even to this day.