Unfortunately, constancy is irrelevant. If a malicious app is using this API to locate you, it can detect when your wallpaper changes and use its app_id (unchanged) plus wallpaper fingerprint (changed) to phone home with "Yo, this user's wallpaper thumprint is now Y."
You can still use as input along with other parts that are accessible without any permissions - for example the android version, device name, IP address...
It isn't special, merely a blue/green colour with mild dithering.
It is a little like looking at the morning sky.
It's on all phones, desktops, etc. And if I were to give it to you? Well. Maybe you're the NSA, maybe I'll be pulled off the street, thrown into a chair, and in a NDA-drugged, mildly confused state, be told "look, it is your desktop, see your familiar backgroud? You should enter your password."
No sir, I will not disclose my background image!! Sneaky! Good try though.