The Color Settings file itself is under your user account (C > Users > your name > app data etc.) together with all the other preference files and user settings, but it's not useful to you. You can't edit that file directly.
What you do is open Photoshop and edit your color settings in Edit > Color Settings. If you can do that, and any changes stick when you next open Photoshop, everything's fine and there's no problem. But don't change anything unless you know why you want to change it. Don't do it because "someone on the internet says so". They're almost certainly wrong.
If your color settings don't stick, you have a folder permissions issue. Your user account doesn't have the necessary privileges. That's an operating system issue that has nothing to do with Photoshop.
And again, if you only have Photoshop any color setting "synchronization" is moot and irrelevant.