Not able to correctly assign color profiles

I have already searched this forum but I can't find an answer to my question.

I frequently work with Illustrator and InDesign, sharing files with a colleague. We have two color profiles we'd like to be set always and for every file, one for RGB, one for CMYK (correct profiles are installed and assigned). We are using Bridge to synchronize the color settings but we face the following issues:

When working on a file and looking at the color settings and assigned profiles, the selected profiles are correct. But when we close Illustrator or InDesing and reopen any existing file, the following messages pop up: "Your current color settings honor CMYK profiles in linked content but profiles were set to be ignored when document was created" followed by/ or "The document has an embedded color profile that does not match the current CMYK working space", showing an embedded profile none of us has ever selected.

• When we select "Use the (wrong) embedded profile" and have look at the color settings, the correct profiles are shown and not the (wrong) one that was allegedly embedded. The color values in the file are correct.
• When we select "Convert to working space", the correct profiles are shown in the color settings, but the color values are wrong.

When we create a new file, we select "embed ICC profile" when saving, but still get these messages every time we reopen the file. The issues stay the same without using bridge for color synchronization.

I'd be extremely thankful if you could help me figuring out what we are doing wrong.