I agree completely with the medical student's concerns. In particular your app asks for information the way a counselor would. See for example this "Record your thoughts..." image from Google Play
https://imgur.com/C5C3A5j. I think there is a very big difference between for example a paper self-help workbook that someone keeps in their own possessions or mental exercises that produce no records and making voice recordings in a networked app that accesses a centralized database stored in the cloud.
Emails are often trivially linked to identity. It doesn't matter that someone could have generated an "anonymous" email address (I've very skeptical that user-provided email addresses would pass muster as de-identified identifiers), if you can use that email to communicate with the person, it's definitely linked to their identity. If they link the email address to their identity elsewhere, the burden is still on you, not the user. You cannot demand that users protect the email address they use to sign up for your service the same way they would protect a social security or medical record number. That's ridiculous.
You should look into obtaining HIPAA/HITECH compliance training perhaps as an unaffiliated learner on https://www.citiprogram.org/ to understand why those of us that work in health care are expressing concern.