Good point thanks for catching that. The receipt itself can name any anonymous identifier like a crypto address. I was just intending to note that the blockchain is essentially a wasteful timestamp server that doesn't seem needed for this application.
As a practical matter, the website has your IP, when you visited, what you looked at etc. So you already have to trust them with your privacy. And there's a question of whether public policy would allow web traffic to be untraceable by default. But certainly the payment processor doesn't need to know things like which websites you visit.