Monologue means a speech from a single (mono) person. The main features of a dramatic monologue are:

• A single narrator (but not the poet) narrates everything.
• One or more other characters are present, but they don’t speak anything directly. We know what they say or do from the narrator’s mouth.
• The text reveals the speaker’s character and temperament.

All these features are present in Vikram Seth’s poem ‘A Doctor’s Journal Entry for August 6, 1945’. So it can be regarded as a dramatic monologue.