My mobile phone company has no idea who I am. It was trivially easy to purchase an iphone, at the Apple store, with cash and then purchase a Verizon MVNO SIM from US Mobile (which I prefer over StraightTalk because they allow tethering).
Neither Apple nor my MVNO nor Verizon knows who I am.
I, personally, am not a customer of any mobile provider.
Certainly my mobile provider could find out if they dug deeply enough - specifically, they could (somehow) check the Visa card I use for payment and see the corporation that that Visa card was issued to and then look up the corporate records and then ... they would know that somebody at X Corp is their customer. Possibly me.
Remember: neither Visa nor Mastercard verifies purchaser name. Only AMEX does this. You can purchase online with your personal or corp Visa/MC and put in "Mickey Mouse" as the name - it will always work unless they are a very, very rare merchant (usually European) that has put the "Verified by Visa" component into their workflow.