wrong purchase date on registration

You need to contact Apple Support by phone:

Update your proof of purchase

An incorrect purchase date cannot be corrected online.

Did you buy the device today or the Applecare? And, was/is it new?

FYI, Apple uses the date shipped for both the regular warranty as well as the Applecare if you purchased it. Since their shipping is usually 2 days, the date will differ by 2 days. If you bought the Applecare today but it is for a device you purchased previously, then the coverage starts on the device purchase date, not today.

If none of that applies, contact Apple; if it has to do with Applecare, ask to be transferred to that division.