Not all the time. If you buy a prepaid card at e.g. Target, they do not collect your SSN or IIRC any identifying information unless you opt to register the card. You might choose the register the card to replace in case of theft.
When I've bought prepaid cards, I either bought them for zero-value or with pre-stored value, and in either case they were not usable until I registered them online. I haven't found any prepaid cards yet that do not follow this pattern. If you have then I would be quite delighted if you shared.
The thing is, big retailers have cameras, a lot of fucking cameras. Better to buy them from the rinky dink mom and pop shops that take cash with no cameras in the shop. They don't care if you are John Doe or George Washington.