Visa/MC and the banks have security measures in place, merchants who follow the process aren't liable for loss from fraudulent cards. Asking for ID provides no additional protection to merchants and to the extent they rely on it instead of established Visa/MC processes it can lessen security.
The info on the ID is the security measures Visa/MC have in place. They allow a merchant to enter info like address or phone number, and their computers will tell the merchant whether or not it matches the address/phone they have on file for that card. When you pay for gas with a credit card and the pump asks you to punch in your zip code, it's not collecting marketing information. It's using the zip code as a (rather flimsy) security measure to protect against someone buying gas with a lost/stolen credit card. Yeah you can ask the customer to recite their address, but any burglar who stole the card from a house or mugger who got their victim's entire wallet would know the address. A photo ID with that info, while fairly easy to fake, requires a bit more effort on the part of the thief.
Credit card security is in the dismal state it's currently in because Visa/MC/Amex have successfully transferred all the damage from fraudulent transactions onto the merchants. Since they lose practically no money to fraud, they have very little incentive to improve security. (The exorbitant interest rates are to cover the cost of credit card holders who default on their debt.) For market forces to work correctly, financial penalties for risks which fail must be linked to financial profits when those same risks succeed. What Visa et al have done is decouple the penalties from the profits (profits go to them, penalties to the merchant), leading to a situation where they are not penalized when the risks they take (poor security) fail. Consequently there is no motivation for them to improve credit card security beyond the laughable state it's currently in.