Maybe your Photoshop is not set to Fogra or has a different setup in your "Edit/Color Settings". There are many options that can throw it off if your InDesign doesn't match.
Otherwise, if you study color management, you'll discover that conversion is tricky. It doesn't just go from RGB to CMYK. First you have to convert CMYK to LAB using the profile and then to RGB. LAB is "device independent" meaning you cannot tell a printer with CMYK inks to "print 24L 189a 70b" (it wants to know how much cyan to lay down). Theoretically, every Lab value should correspond to a very exact color in the physical universe, in very exact lighting conditions. But unfortunately the LAB conversion isn't perfect, so if you convert CMYK back to RGB, you get a different value. At least the conversion in one direction will always come out the same (when you use the same profile and settings). The French are always researching further into color. Maybe one day they will figure out a better model than Lab.
FYI, it is best to set your master color settings in Adobe Bridge and save them as a preset, which then makes all Adobe programs use the same Color Settings.
Lastly, if you are on a cmyk project, your InDesign should definitely be in cmyk, but your Photoshop does not have to be tethered to such a small gamut. you can use Adobe RGB 1998 or sRGB, so your overlays and special effects will look bright and gorgeous, and InDesign will do what it can to convert that bright and gorgeousness to CMYK, according to the profile you chose.