Seriously, how can your cat get his own credit cards if he has no SSN? Credit card companies require the SSN of the primary cardholder. Moreover, any application filed "by" your cat would be fraudulent, since your cat can't legally sign the application. My answers to your questions would therefore be, "No, not legally." Of course, IANAL, I have no idea what the CC company may or may not catch, and I am neither encouraging nor discouraging this idea.