Well, both monitors shouldn't use the same profile. There should be a custom profile for each monitor. But it also seems like the calibration is wildly off on one of these displays (not the same thing as the profile).
There is a tool that will fix both problems: a calibrator. It will first calibrate the monitor to a certain white point and neutral color balance relative to that white point. Then it will write a monitor profile that describes how the monitor reproduces color in that calibrated state.
Run the calibration and profiling on both monitors, one after the other.
Aside from that, the behavior you describe is normal. Photoshop uses the monitor profile it gets from the operating system for that monitor. When you drag over, it switches to the other profile when you cross the mid line and drop.
Here's a good one: