I'm confused. I got a new work email. I updated the email associated with my named user account using ArcGIS Online, and then yes, verified it. But my profile still displays [my name | old email] in the yellow box in the lower left under Users. Ditto for in my profile on AGOL, etc. Note that the account I am using now is my personal account. I cannot find a way to change that?? Seems like this should have automatically updated and/or have a simple solution.
Is the ArcGIS Online site hosted by your work?
It's hard to tell from the screenshot - but if it's a hosted site, depending on the login your system/enterprise admins might have to update it! In the example of the site we have, we have SAML/OATH login. So when we create accounts, it's tied directly to our system and not really ArcGIS Online. Even as an ArcGIS admin, if I tried to change my email to my personal email, it would either overwrite the settings or probably lock me out. So if it is hosted, I would definitely check with a system admin!
Not sure if this is the case, but that could be causing the problem.
Good luck!