Post a screen shot of each application's Color Settings. Use the Advanced view. If you are getting the color you want out of Illustrator, why not end it there? Is this for web view? Wherever the file goes, there may be a shift because of different user settings and browser capabilities.
I'm going to take a very educated guess that your problem is the jpeg engine in Illustrator. Jpegs exported from Illustrator do exactly what you are describing, but if you export a tiff or a pdf, or do a Save As and save as a tiff, there is no numerical change to your pixel values. Saving a jpeg from a tiff brought over from Illustrator results in an almost identical values, with the final result being 0,0,152 rather than 153. That is close enough to not be able to see.
Inconsistency between or among applications in the artificial "suites" should come as no surprise.
The "suite" concept is a fabrication of Adobe marketing and bean-counting types. The engineering teams are totally independent of each other, they are not only in different buildings but in different cities and states of the American Union, even in different countries and different continents.
The fact that they have little if any communication among them is highlighted by requests occasionally made in these forums by top Adobe engineers to let the other teams know when there are problems in one application that impact our workflow in another one.
First, check to make sure that the RGB working colorspace is the same for both Illustrator and Photoshop.
When exporting the jpeg from Illustrator make sure the 'embed' color profile is selected, and it is the color profile that is required for your client. If this is for web use, then sRGB is your best choice.
When you open the exported image in Photoshop, make sure that the color settings for Photoshop are set to 'Preserve Embedded Profiles'. After you open the image in Photoshop, you can double check by going to Edit > Convert to Profile... The source profile should match the embedded exported profile from Illustrator. If these match, your RGB color values from Illustrator and Photoshop should also match.
If the source profile does not match, cancel out of the 'Convert to Profile" and go to 'Assign Profile". Assign the profile that you embedded when exporting from Illustrator. Now Photoshop should have the same values as Illustrator.
Good advice, but it seems, the OP is already familiar with synchronized color settings:
I have activated Adobe CC Color Manegement via Adobe Bridge (option "Europe, Universal 3"), which should produce consistent results in all Adobe applications.
And yes, Europe Universal contains RGB = sRGB.
When I open the file in Photoshop, the color is RGB 1/0154.
Probably 1/0/154, which is near to to the source 0/0/153.
No reason to worry about, the colors are perceptually identical (or very near to each other, as already mentioned).
JPEG compression implementation contains plenty degrees of freedom, therefore such a difference is not unusual.
Best regards --Gernot Hoffmann