--this must be a new issue. --
I think the problem may have started after the latest Windows update. The computer rebooted overnight some days ago. The problem started around that time, although I can't be certain it's due to that update (such seems likely though).
Since you are willing to test could you try same test without checking remember me. That options saves a cookie and OP is trying to get it to use saved Password.
On all 3, Firefox, palemoon and Chrome, if I sign-in to eBay, do not check "Stay signed in" option on eBay login page and close the browser while signed in, I'm asked to sign-in again. So the cookie works, as expected, on all 3.
I notice, if I check 'Stay signed in' on Edge and close without signing out, upon Edge reopen, the box is Unchecked, so that may explain why sign-in required every browser session.
Just spoke with eBay tech support. They have had several calls about this same issue (Edge won't hold eBay passwords). EBay launched a new sign-in page recently, and this is when the problem started. Other browsers hold user passwords but not Edge. This was confirmed here by mrgeek too (thanks!). I was told that eBay site engineers are working on the problem and of course hope to resolve it soon; hopefully within a week but there is no way to tell, no set time frame for the repair. EBay recommended using another browser as a temporary solution, but as we know, not everyone wants to use a different browser for one website. I'll probably just visit less often and type in the password while waiting for the fix.
While I understand your desire to save passwords in Edge I would be following up on that exposure. If indeed you can run a simple program to expose your passwords I would not be using.
Password managers are an alternative way to handle.
If you track ebay's performance if you wouldn't mind posting back when it is fixed.
Interestingly and unexpectedly, eBay phoned me within a few hours of my earlier call, to reconfirm they are working on the problem. They also said that their site engineers created a case for me and for the others who have contacted them to report the same problem with Edge remembering eBay passwords. In any case, it's good to know they are working on finding a solution for their new sign-in page; once remedied, hopefully it can solve the issue for all Edge users.
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