Depends on where you are!
Linkage to follow, expect updates:
Visa and MC really don't want the customer slowed down or pissed off. They have rules preventing merchants from requiring additional ID for purchases as part of the regular process. Returns, by contrast, _can_ require state ID.
Basically, everything people did when you used to pay by check seems to be Not Allowed, and maybe that's part of why no one takes checks any more. However, this isn't necessarily the kind of privacy rules I was looking for.
But that's more about borrowing/lending/etc. financial institution stuff, not going out and buying something from a store, or buying something from an online shop that is then mailed to you.
Another way to think about this is not a right to privacy but a right to control publicity associated with one's name.
Chuck Yeager sued a wireless company for using his name in a press release and won (I have no idea what happened on appeal, and honestly, the amount won didn't look like it justified the effort put into the suit), and also sued Virgin America for something similar.
This is a law firm warning you that this kind of thing can happen:
They mention this lawsuit, which was settled:
It's unclear what the total was from that, but there was a lot of money from a related case, that he then donated:
However, all the right to publicity laws are, er, trumped by first amendment rights. So while a shop that said this:
"Famalicious McCelebrity Shopped at FunFunSexyTimes!!!!! Order Your Sex Toy from FunFunSexyTimes Today!!!"
would probably lose in court should Famalicious McCelebrity choose to sue FunFunSexyTimes, I suspect that "Hey, It Is Political Speech And Thus Protected", if what FunFunSexyTimes said was actually this:
"Famalicious McCelebrity Shopped at Our Store and We Donated the Proceeds to support SomeCauseOrOther".
Especially if there was any reason to believe that Famalicious McCelebrity opposed SomeCauseOrOther.
I am trying to figure out what I think of all this. But the fact that this is not something that generally arises is all by itself kind of interesting. I mean, we've had lawsuits about whether the guv'mint can take a gander at your library record and what kind of warrant they need to do so. There's all kinds of stuff that stores can't ask the purchaser for. But boy, if the store knows who you are, and they are not actually trying to make money off you in an apolitical way -- like, if they wanted to share with the general public your shoe size or how you liked your burger prepared or that you really hate coleslaw -- there's really not much stopping them.
If you find evidence to the contrary, please share! I'll probably retain some interest in this for a while.