I thought some were anonymous when paid with cash, however some of those require that they are activated using a web browser and they request additional info, some requiring physical address and SS that match some database in order to use them.
I have also seen some businesses refuse to take a prepaid cash card, some saying nothing,
and others saying that they can not process such a card as recurring billing is blocked by the issuer or something.
I think there is some distinction at the card kiosks re-loadable vs non-relaodable making a difference with some of these issues.
Of course using a loyalty card during purchase or using a debit card to buy one adds personal info into multiple databases I would assume.