The Serial Monitor window does not allow to do what one expects.
Reference: the Output window: a right-click allows to copy/copy all/ clear and other. Also, a CTRL-A marks everything in Output, and CTRL-C allows to copy it.
Within Serial Monitor (SM):
a right-click does nothing.
a CTRL-A marks nothing in the SM, it rather marks everything in the active code window
a CTRL-C copies everything in the SM currently marked, PROVIDED you can enter CTRL-C fast enough before the next update of SM, because that unmarks everything.
This makes it hard to impossible to use the SM text for a deeper analysis.
What should happen: SM should behave as the same as Output window
Edit: Output has an option to "Turn Auto Scrolling Off". SM needs the same option
And while replacing the monitor, please also look for consistency in icons, symbols and other visuals:
In "Output" the Autoscrolling ON/OFF is a padlock in locked/unlocked state.
In "Serial Monitor" the same function has "looks like two downward pointing carats". This reminds more of a drop-down something.
I agree fully. Because the Arduino IDE is intended to help make embedded systems accessible to everyone, this sort of thing is far more important than it is for IDEs targeted solely to professionals. I think it's likely there will be a complete reworking of the Serial Monitor UI during https://github.com/arduino/arduino-ide/issues/105.