Safe Mode red sign disappears after login
I noticed the Safe Mode sign in Red color, displayed towards the top right corner on the login screen, however, after login, the sign disappeared.
  
  I thought that once you are in Safe Mode, this will last until you restart your machine normally.


The Safe Mode text in Red color is shown towards the top right corner only on the login screen and disappears once you login. This is by design. This doesn't mean that your computer has exited the Safe Mode. It's still running in Safe Mode.




  I wonder why my Mac automatically exited from the Safe Mode (or not)?


This is not the case. Your computer is still running in Safe Mode after you login from the screen indicating Safe Mode boot.

To further verify if your computer is running in Safe Mode (or not), go to Apple Menu → About This Mac → System Report..., and in the left sidebar, click on Software. In the right pane under System Software Overview, look for Boot Mode item and check its value.

A value Safe would indicate that the computer is running in Safe Mode.



Whereas a value of Normal would indicate that the computer is running in Normal Mode.