Credit cards are trivially traceable to your legal identity, since anti-money-laundering and know-your-customer laws require that credit card companies keep this information. The government can subpoena this information just as easily as they could with pictures of your face or ID.

How do you prove the person typing in the credit card details is the same person who owns the card?

I know I've read stories of kids taking cards to purchase games or other things online numerous times over the last 20+ years.