ArcMap maintains the stored credentials to ArcGIS Online despite closing the program (and killing the Connection in the taskbar).
How does ArcMap store these credentials?
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Disclaimer: It should go without saying, don't mess with your registry if you don't know what you're doing! I take no responsibility.
It's stored in the Windows Registry:
Here is a screenshot of part of mine.
To test, I made sure I was signed into my ArcGIS Online account and closed ArcMap. I also had to check that the Connection Manager also closed.
I renamed the registry key to
ArcGIS Onlinexx, closed REGEDIT and started ArcMap. I was no longer signed in and the login details were gone.
I then closed ArcMap again, renamed the key back and restarted ArcMap. It signed in automatically again.
To take it a step further (I got curious!):
If I was needing to switch accounts often I could see benefit in a batch file to populate the correct registry key for whichever account I want to be signed in on.
Note: the login credentials are encoded in the Registry - I cannot see the actual Username or Password.