Amazon would gain big profits from selling user's information and behavior to marketers and anyone who needs such data
and they could let people buy it anonymously but not true/pure anonymous as the buyer would hope
in the end they will also sell all the buyer's data to their (buyer) competing companies
in the same business sector The thing is, it's just impossible to buy anonymously from big retailers like Amazon, they wouldn't be able to deal with the on-chain volume, it would be a disaster, so they would need to resort to the usual third parties like Bitpay and other platforms which by default are compromised.
Then we would have a LN node which may or not be viable anonymously.
Of course big block sizes aren't also the solution.
Basically there's no way around this, so if Amazon said you could buy anonymously, you have to assume they are lying. Anonymity is never possible unless peer to peer, not peer to corporation.