You would be correct about Photoshop using the profile of only one monitor. Even when you have a single monitor, images can have a color cast when open in Photoshop due to how Photoshop handles color. It's not that Photoshop is wrong, it's just how the color settings are configured.
The first thing you have going against you is that you're using two monitors made by different companies. Even if they were made by the same company, the chances of getting two monitors to exactly match each other aren't great. So, when you color calibrate them you're calibrating them based on where their current color settings are. If the color settings aren't similar to begin with, such as one monitor is displaying more reds and the other more greens, then your calibrating how the video card handles color sent to the monitors. When Photoshop launches it uses the color profile of one monitor, so when you move a panel from the monitor that Photoshop is using the calibration of, the image is going to look wrong on the other monitor because Photoshop is displaying the image how it was told to show it, thus the image doesn't look correct on the other monitor.
The only way you're going to get the monitors closer in appearance is to purge the current calibration and manually get the two monitors to be somewhat similar in appearance using the monitor's color settings. Then when you do a proper calibration you'll get a closer match when using Photoshop and move the images between the two monitors.