OS: macOS Ventura 13.6.1
Software: Illustrator CC 2023, Indesign CC 2023, Photoshop CC 2024

Steps to Reproduce:

Step 1:
Set a unified color profile across all CC applications using Bridge (In this case, Adobe RGB + Japan Color 2011 Coated were used).

Step 2:
Create an image in Photoshop in RGB mode with a solid cyan color (#00ffff), embed the color profile, and save it.

Step 3:
Place the image created in Photoshop into Indesign and check the values when converted to CMYK in Overprint Preview.

Step 4:
Repeat the process from step 3 in Illustrator.

Expected Result:
Both Indesign and Illustrator should output the RGB image in CMYK with the same values (in this case, the calculation would be C69, Y31).

Actual Result:
Indesign shows the expected result (C69, Y31).
However, Illustrator outputs different values (C64, Y25).

Additional Observations:

This discrepancy is apparent both on-screen and in direct printouts or when exported to PDF (X1a).
If the color profile is not embedded in the PSD, Illustrator outputs the image as C69, Y31.
The color profile matching method (Perceptual, Relative Colorimetric, etc.) is set to Perceptual, but the incorrect results in Illustrator match the values of a Relative Colorimetric conversion.
This suggests that Illustrator is not displaying placed images correctly according to the set configuration.
I am sharing the files used in this test for reference.
https://shared-assets.adobe.com/link/1e706170-630c-4917-52ab-c58a9955dfc8