Your monitor can't display a higher resolution just because the computer tells it to. It only has so many pixels. Setting it at a resolution higher than what it supports will do nothing but lower your image quality by making the screen a bit fuzzy.
If you want your resolution to be 1920x1080, buy a new monitor. You can't trick your old one into doing it.
Edit for clarification:
What's happening is your computer is sending a display signal to the monitor that represents 1080p. Your monitor, being fairly smart, figures out that it can't display that 1080p image. Instead, it will convert it to 1600x900. What you're seeing is exactly the same resolution as before, it's just been garbled slightly in the conversion (thus the fuzziness).
Games, however, work a bit differently. They do not take their resolution cues from Windows, and instead handle it all themselves. This is why you can't select the resolution in games, they understand that your monitor can't display it.