This doesn't make any sense. First off, when a store is asking you to apply for a credit card they indicate it. Secondly, when you are applying for a credit card you have to provide your DL, SSN, address, income, etc. You had to have provided this information to further apply for the credit card. You also would have had to sign electronically that you were applying for the credit card.
The other item that doesn't make sense is if you were using your "other credit card" did you not give it to the clerk or swipe it? You had to have known the charges didn't go onto your credit card you wanted to use.
The receipt would also have the last four digits of the credit card used and further more you would have received an approval receipt for the credit card.
Doesn't make sense at all.....