The CDC card is very basic. It's not numbered, has no QR code, and there's no central registry. It's also printed on ordinary card stock, with no watermark or other security measures. So it's not really a Big Brother device.
Vaccinations are in fact officially registered in the U.S., but they're registered at the state level. The states though use different computer systems and have different methods for getting a copy of your official vaccination record.
Point is, when you flash your CDC card alone it's impossible to verify it. But if you come under suspicion it is possible to determine whether you in fact received the vaccine.
https://covidresponse.wisc.edu/content/ … e-card.jpg