As the title say, I've got this issue with Illustrator. I'm working on a CMYK document and if I copy and paste an object to another CMYK document, the colors of the object changes in an "unsaturated" version of the same color.
I have written a lot in comments, now I think have enough info to attempt an answer.
Spot colours are not CMYK colours. They are a single solid coloured ink - mixed from other colours according to a formula guide created by Pantone.
If you adjust the spot colours in Illustrator using the colour swatch options by moving the sliders, you are only changing the display colour, the colour isn't really changing. When you copy and paste the colours you changed into a new document, they are reverting to the colour swatch book (the Book colour) - because you can't edit spot colours like that.
Perhaps it's time to familiarise yourself with CMYK colours versus Spot Colours, what they are and how they differ.