Displaying both RGB and CMYK in the same document

Also, Spot Colors can behave differently, the explanation at the end of this thread might help:

https://community.adobe.com/t5/indesign/indesign-rendering-illustrator-linked-graphics-differently-across-pages/td-p/10873261?page=1

InDesign lets you mix RGB and CMYK colors or objects on the same page. An RGB color might be displayed in the document’s assigned CMYK space depending on your settings.

Here I have an RGB (sRGB) and CMYK (ISO Coated v2) process color on a page with no transparent objects, and Overprint Preview turned off

If I turn on Overprint, the RGB object remains as RGB, but gets displayed in the document’s assigned CMYK space:

InDesign has a Transparency Blend Space to handle pages with mixed color spaces, and any transparency on the spread. Here I have my Transparency Blend Space set to CMYK, so if I add a transparent object to the spread the RGB object displays in the blend space. If your placed AI logo has any transparency, the blend space would be invoked:

If I set my blend space to Document RGB, the RGB object remains unchanged even with transparency on the spread:

There is no such thing as RGB or CMYK "as it is".

The numbers don't mean anything until you define the color spaces in terms of icc profiles. sRGB, Adobe RGB, ProPhoto, DCI-P3 are the standard RGB working color spaces. Each of these specifies a certain set of numbers for any given color. Inversely, any given color yields a different set of numbers in each of these color spaces.

CMYK is even more relative. An ideal synthetic CMYK color space doesn't exist. A CMYK profile represents a set of printing conditions - an offset press using certain inks on certain papers. Presses are calibrated to different standards, and standards vary in different regions of the world. For instance - the Photoshop default is US Web Coated (SWOP), which is fairly widely used in North and South America, but it will get you nowhere in Europe. Other standards apply there.

In short, you need to take a much more systematic approach to this. For RGB you can assume sRGB as a fairly safe bet, but for CMYK you need to look at what standards commonly apply in your area.

The one thing you absolutely must not do, is proof to your display. That turns off all color management and throws everything to the wind.