Sometimes I do use several bonus cards, when there's some significant benefit (say, sale-out only for bonus-card owners). Guilty that. But I do not use them often and of course I don't use my real name when signing up for it. Or if I do — only when I use my CC anyway, which is rare.
Surely if you use your CC one time with this card they have your real details. After that every cash purchase where you use this card is tied back to you. Not hard to believe they do this and you should assume the worst.
