artificial "suites"
I don't agree with that; some of these applications are very tightly integrated with a seamless cross-app workflow.
Anyway: In itself, there's no particular reason to have the apps "synchronized". Requirements are different in the various apps, and my color settings are deliberately unsynchronized. But until you understand more about what the different color settings do, it's a simple safeguard against surprises.
In Photoshop, working on RGB (or CMYK) raster image material, the all-important thing is to have color management policies set to "preserve embedded profiles". Any other setting is trouble waiting to happen. You may also want to work in a large space like ProPhoto to have maximum editing headroom.
But as soon as you get into CMYK and/or vector territory with ID and Ai, you don't want that because it can lead to a number of other problems on press. For CMYK you'll want to ignore profiles and preserve numbers, then reassign the correct profile for output. Using ProPhoto for RGB vector is also totally pointless. Just mentioning this to put this into context.
If you do want to synchronize, you do it in Bridge as station two said. But all it means is having the same settings in each app's color settings, and you can do that manually as well.