If you have a subscription, then your Adobe ID is all you need.
If you have a perpetual license, then you need to get your serial number -- either from the box the software came in, or from your account information on adobe.com.
As for reinstalling - just download the installer and enter your serial number or Adobe ID.
I don't see any registered products under your account looking you with the email address you've used to sign in here. Is it possible it's under a different ID? If you didn't register your product, then we have no way of looking up the serial # if you lost it.