How to restart using an app without it accessing older data? An app that I tried out a couple years ago somehow keeps identifying my old trial data (it required a signup with email, but now I'm trying to signup using a different email), even if I delete the app, reset the advertising identifier, and reinstall the app. The app did not require connecting with any third party data, such as signing in with a google account.

Is there any way that I can "anonymize" my device without having to completely reset it?
It depends on how the app was made. The way you describe it, it sounds like the app uses Apple's DeviceCheck to ensure that it can only be trialed once.

If that is the case, then you cannot as a user reset the data - not even by completely resetting your phone.