Why don’t stores require proof of identity when using a CC? I know. They don’t really care as long as they get paid...but...
It is against the contract they sign with MasterCard, Visa or any other credit card company. The stores are the ones who get stuck with fraudulent charges, but the CC companies won't let them take any stronger anti-fraud actions than checking the signature and getting confirmation from the CC company.
Some people have "Check ID" on their signature line, but even that isn't allowed under the CC agreements. However the only place I've heard of which doesn't allow that and then check ID is the post office.