I run Lightroom on one monitor and Photoshop on a second monitor. Photoshop opens using the color profile of monitor one. Therefore, the colors are not correct in Photoshop. I want Photoshop to use the profile of the second monitor and Lightroom to continue to use the profile of the primary monitor. Is this possible?
So what happens if you switch display assignments in the OS? Are they both connected to the same video card? What video card, specifically? What monitors, specifically? What calibrator(s) are you using?
Generally, Photoshop uses whatever monitor profile it gets from the OS for each monitor. The profile is loaded when Photoshop starts, and there's nothing in Photoshop (on user level) that determines which monitor profile to use.
That's not to say there isn't a Photoshop bug somewhere, but we have seen a lot of these cases with iMacs/MBPs recently, to the point where it's a very conspicuous common factor.
If this happens to be a BenQ monitor - I'm just guessing here, since you don't give any information - the BenQ software is known to be very buggy and cause unpredictable behavior, so that could also be it.