In a dramatic monologue, the presence of a silent listener is always important in bringing out the key character traits of the speaker. It really depends a lot on who the listener is and how fluent and expressive the narrator can be before that listener.
 
In Nissim Ezekiel’s poem ‘The Professor’, the presence of the student helps the narrator, the professor to freely talk about himself and his family. As a professor, he holds the upper hand in terms of social recognition and confidence while he is talking to his student. It only helps us know his views and thoughts on various social matters. This helps in revealing his stereotype ideas that most Indians do have. Thus, presenting the student rather than anyone else serves the poet’s purpose well.