CMYK colours changing

I'm not entirely sure, but I believe what you are observing is a subtlety of the interaction between Mathematica and your operating system. You use a color-chooser dialog (Format -> Font Color -> Palette), which definitely interacts with your OS. The reason for my assumption is, that on my OS I can pick not only colors in the Mathematica notebooks but from the entire screen.

When I use now this color to draw another rectangle, I get the following image

Clearly, this is not the same color.

I think the following happens: You use the magnifier to pick a color, but instead of picking the CMYK color, you simply pick the RGB pixel value of the screen. Then you can display the corresponding CMYK values again, but somewhere in this conversion your original values are lost.

When you export the graphic directly to EPS, then the Mathematica Kernel takes care that the color values are kept.

Although it is true you are observing an interaction between operating systems. The reason for that is due to the choice of CMYK values. The appearance of most CMYK values are dependent on the gamut of the CMYK color model used in the software where those CMYK values were chosen. You mentioned that those CMYK values changed appearance when you moved to Illustrator on a PC from Mathematica on a Mac. The default CMYK Space in Illustrator can be checked by using "control+Shift+K" or command+shift+K on a Mac. Having one application that uses a generic Quartz CMYK space, and Adobe Illustrator which uses a defined CMYK device space, is the reason for the visual differences and the measurable differences in RGB measurements from each color. To get around this issue you can assign an ICC profile to the image you create in Mathematica, then the color will not change so long as the profile is maintained accross the various different softwares and OS's