First, you are correct, VISA/MasterCard do not require that the name be included when a card is submitted for authorization. (Note that even if they did this would not eliminate fraud. When the crooks hack a card terminal or processor, they steal all of the information that is transmitted to make the counterfeit card. If the name were transmitted as well, they would steal that too, and have the name on the card be correct.)
Also understand how counterfeit cards are made. I buy a VISA/MC gift card at the store, use the money, and the wipe the mag stripe and re-encode it with the stolen information. There is no name on these cards so I simply sign whatever name I am using (probably fake ID to go with it) obtain a cash advance from the account at your institution, and my signature matches the back panel perfectly.
The short answer is, "No" you have no chargeback rights and are on the hook for the loss. Make sure to report losses to MasterCard's SAFE or VISA's FRS systems. In the event a data breach is found to be responsible, and a merchant is found to not be PCI Compliance, VISA/MC could levy fines against that merchant and offer some reimbursement to those who took losses.
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