You're "required" to give your phone number, e-mail address, and mailing address to sign up for a discount card or delivery any most other grocery stores, no? It's up to you if you want to give them real information, although it will probably make it more difficult to recover a lost card or order in the future if you did.
I'd imagine that Amazon also probably gives an option to unsubscribe from promotional text messages if you give them your mobile number. If not, you should be bitching about their privacy policy on Twitter or Facebook. Nobody from Amazon is going to see your complaint here.