I work on crap like this, and it depends. Yeah, diagnostics are done in the car - the main ones, that is. But for example BMW collects data from all their cars - they’re able to do some big data analysis. I wouldn’t be surprised if some of the diagnostic info this app provided is an inference - your car has software version 4.3.21, and all cars on that sw version have experienced a certain bug at 200k km, so it’s time to go to the dealership or get a remote update or something. It could be done.
Most likely though, they’re just taking the personal data from your car and showing it to you. You know - after also saving it for themselves.