I don’t think activating the laptop is related to the desktop needing activated. My guess it the desktop needing activated is due to a PS or plugin update. Sometimes when Photosohp updates, it loses the activation files. I have reported this to Adobe a few times but they have not fixed this problem as it sometimes still happens.
Your key is a 2 computer key.
You can view or clear activations on this link. However, it only shows 1 computer being activated and the other slot available. So you just need to activate the desktop and they will both be activated.