How do I identify and handle cloud storage spam on iPhone?

There are many scam messages that will make that claim. Apple does provide an email that your iCloud storage is full, but that will address you by your actual name such as "Dear John Smith". The scam emails are always addressed with something like Dear Customer or a generic term that can be sent to thousands of people.

The one thing you want to be aware of is that there is a difference in phone storage and iCloud storage. You said your "phone has a wad of storage", and that has nothing to do with your iCloud storage. Review the difference in iCloud storage and device storage here, as well as checking each one.

What's the difference between device storage and iCloud storage? - Apple Support