Do you need a photo ID to get a credit card?

No, you do not (with a couple of exceptions). The merchant agreements for each of the credit card companies strictly prohibit a merchant from refusing a credit card if you do not show an ID. Note that the merchant can ask, but you can say no and he cannot turn you down.

The exceptions are if your card is not signed or certain cash transactions.

More to the point, you should not show a merchant your ID. This opens you up to identity theft since you just showed a stranger (probably minimum wage) not only your credit card information, but a great deal of other personal information including date of birth, address, legal name, drivers license number and other information.