Move feature tree to 2nd monitor?

To the best of my knowledge, no.

The feature manager is not model specific. The feature tree is, so it has to be tied to the window with the model.

This probably doesn't make sense until you open up multiple windows at once (not full screen) inside SolidWorks. Open up two part files, and go to Window>Tile Horizontally.  Note that each feature tree is tied to each window. The feature manager isn't, so it can be undocked and put wherever, including on a different monitor.