The only really possible way to buy btc staying anonymous is by cash.
If you buy it with some kind of online payment, the selling party always knows your name or account details.
Or he can simply buy prepaid virtual credit card's most of the providers that i know offer you a 2.500$ yearly spending limit and you not need any from of id card, it would be hard again to buy but it's a a big chance that this is a solution, let's say you load the vcc for 100$ and you can simply swap them for amazon gift card or other online gifts the excgange rate is terrible but again an option to stay anonymous