I got a hold of Fritzing and find it pretty cool so far but there is a disadvantage to the fact that the serial monitor has no way of sending things to the board like the Arduino IDE does, so include this in the next update please!
Welcome aboard! For the serial monitor you likely need to make an enhancement request on the Fritzing-app page at github. It may be a limitation in the Arduino IDE rather than in Fritzing though (but I don’t know for sure!) because Fritzing is using the Arduino IDE API to do the work.
I believe that Fritzing is using the (terminal) command line to compile and upload sketches to Arduino boards. It does not actually use the IDE portion. Which is where the serial monitor lives. If this was to be implemented, I doubt it would use the Arduino IDE. A clone of that functionality would be possible, but the priority would be quite low. Too many other features and bugs to work on that are part of the core Fritzing functionality.