You probably have a different CMYK working space set in the two versions.
I didn't think to check those. Can the working spaces be file specific? Because if I go to Edit>Colour settings in InDesign on both files they display the same information. Illustrator displays rich blacks as expected and it has the same colour space settings as InDesign (Europe General Purpose 3) as well.
EDIT: Turns out I was in the wrong screen, Edit>Assign Profiles showed me that the CMYK settings were indeed different, so that has now been fixed.
In answer to the question can profiles be documetn specific, indeed thay are. Waht you have set in your color settings is the default condition, but you can change it for any document at any time.
Can the working spaces be file specific?
Unless your CMYK color management policy is set to Off, profile assignments are always document specific—the Color Settings' Working CMYK profile is not used.
That rich black comes out as brown.
If you make the rich black with 100% black and enough CMY (70|60|60|100), its color will preview the same with any profile.