A question also posed at Is there a word for answering a question with a question?, over at Stack Exchange.
Maieutics and the Socratic method are not it. That’s Socrates’ use of questions to encourage someone to rethink their premises; Socrates wasn’t ELIZA.
The answer with a long list of rhetorical figures, posed in the form of a bunch of questions, is inaccurate too: those rhetorical figures cited are just questions, as you can check at https://rhetfig.appspot.com/list.
The only reputable-looking answer is counterquestion, which is on Wiktionary (counterquestion – Wiktionary), and in the OED, dating from 1864: