I don’t recall seeing a circle-exclamation-point icon for a bad serial number, but I’ve been on the cloud version of LR for over a year so maybe I’m forgetting or maybe it’s changed or maybe what it signifies is different than a typo-red-X symbol. If you purposely mistype one of the digits do you see a red-X, so the circle-exclamation-point actually means something else?
Anyone who’s used Adobe Chat Support wouldn’t think you were making it up. It’s excruciating unless it fits into a well-known script that they can match the situation with early on.
What information did the chat session reveal about your serial number?
Is the serial number something that you or your work purchased for you as an individual license or is it some sort of team or enterprise or other group licensing?
Has the serial number ever been associated with a different AdobeID?