How to deal with student's feedback about tutor
No, don't mention the student by name when speaking with the tutors. Yes, the professor should be made aware that there are issues, but that you will work to resolve them. The professor might need/want the name.
You may need to speak to the individual tutor to get their feedback on why they are doing what they do. There may be a good explanation that is actually student focused. Or not. But you need this feedback.
However, you need to spend some time with your group of tutors to make sure that they treat students in a uniform way. All of them should use the same criteria and the same policies. If that isn't happening then some students are being advantaged or disadvantaged over others. This, I think, is the most important element here.
But if you think a reprimand for the tutor is necessary, the professor needs to be involved in that as well.
And, it may also be necessary to reassure the student in some way that their concerns are being addressed. How you do that should also involve advice from the professor.