Old copy of Elements won't open, how do i update?

I don't think Elements 15 would have worked if you didn't register it.  Make sure you are signing into the Adobe account with the email that you may have been using ~8 years ago when you first purchased the program?

Did you follow the link I posted which takes you to the Products page of your Adobe.com account?  If you sign in with the credentials you have used to sign into Elements 15, your serial number should be there even if you purchased from Amazon with a redemption code. 

You can start a chat session by clicking on the blue Contact Us button under the Help & Support menu at Adobe.com. Or you can contact Support by telephone using the numbers listed here: https://helpx.adobe.com/contact/phone.html. Hours of operation outside of the United States are limited. See the phone link for details.

Of course, there have been many improvements in Elements since version 15, so it may well be time for you to take advantage of the Black Friday sale.

I'm not sure where hatstead got the idea that you are not the initial purchaser???

Your serial number and a download link to your registered software is found on the Products page of your Adobe.com account. 

Each version of Elements is a separate program and normally you would have a prior owner's "upgrade" discount for a newer version purchase.  But, as hatstead says, Adobe is currently selling Elements 2023 for a price lower than the normal discount.  The Black Friday sale ends on November 29 in the US.

You can download most non-subscription apps if you purchased them from Adobe.com or registered them, but since you don't have the serial number and are not the intial purchaser, that is not a viable option.

I noted that PSEv.23 is available currently for $59.99!! I have no commercial connection to Adobe.