This is ASP.NET built-in functionality, as MonoX is using ASP.NET Forms authentication by default you get the Session Id saved in the cookie, so after you change the username for admin that Session Id can't be validated anymore and that is the reason you get the security exception.

Regards   

Why should the admin experience such a behaviour though? It should show a prompt to the admin that since they renamed the account they're currently logged in with, they will be immediately logged out and they should log in again with the new account name or other admin account. Pressing OK would log them out

A related issue is when admin is in the admin console, then logs out, then logs in as a user to try the site and they get errors about illegal access etc.
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</br>When user logs in with non-admin account and the login screen sees that it is supposed to take them to admin console they could be redirected to the home page instead.