opened your document. One can see the issue already in InDesign.
 Never use swatch [Registration] for a fill color! Never!
You applied swatch [Registration] to a rectangle in the background.
 Your placed artwork from Adobe Illustrator is missing so I cannot tell what's the right "black" to apply on InDesign elements to match that "black". Ideally it would be in the same color mode and applied would be the same color values used in Illustrator. But this suggestion comes with a warning. See below:
If the pages are for printing on a printing press with CMYK inks you better change the fill color not only in InDesign for all the background elements, but also within Illustrator. This would require Illustrator files in CMYK color mode where ink Black is 100% and the other inks would form a neutral gray when printed together. To get the right percentage values for that neutral gray from inks Cyan, Magenta and Yellow, you absolutely have to know the printing conditions, the target could be e.g. "ISO Coated v2 300%" or "PSO Coated v3" here in Germany or something else totally.
Your document CMYK is set to U.S.Web Coated (SWOP) v2; if that really describes the printing condition accurately the absolute maximum of "rich black" could be:
71 % Cyan
65 % Magenta
100 % Black
That is 300 % Ink in total.
But I would use not that much of Cyan, Magenta and Yellow ink, because it seems that there is small type in white on that black filled area.
But first let me know how this should be printed or if it should be printed at all.