Second round interviews are sometimes more in-depth than the first round. In many cases, the first round or screening interview is with HR, and has some general questions (tell me about yourself, what are your strengths and weaknesses etc.). The second round is usually with the hiring manager and/or the team you’ll be working with.
Second round interviews with the hiring manager give you a chance to ask the supervisor questions like “what is your management style”? or “what would you like to see happen in the first six months if I were hired?” or “what would success in this job look like for you?” I think all of those are good questions to ask if the person interviewing you would be your future supervisor.
You may get also more specific questions from the hiring manager than you would from HR. If the job has any technical component, the hiring manager might ask questions to assess your proficiency in technical areas of the job. Usually the hiring manager will have more specific ideas about what the job entails, and might ask you to say what you would do on the first few months of the job. You can prepare the same way you would for your first round, just think more deeply about what you would do if hired.