Is there anyway to remove this ugly and unnecessary icon?
I remove 'indicator session' from startup programs but has no effect.
Remove the indicator-session package from your system - the indicator will be gone for good after the next login.
sudo apt remove indicator-session
As a side note - it's always been desired to appear in the top-right, like pre-Unity times, which has finally been fixed in 19.10 and onwards.
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Another way to get what you want without removing anything is to just disable the service and reboot.
pkexec to get temporary admin rights, navigate to /usr/lib/x86_64-linux-gnu/indicator-session/ and rename indicator-session-service to indicator-session-service.disabled. Close super user and reboot.