The page searchdomaincheck returns your preferred Google domain. For me, visiting it returns
https://www.google.com/ normally, but
https://www.google.com.py/ in a temporary profile and incognito mode.
This is its main function, although I fail to see why it is needed if you're not using the Google search engine. To make this possible, Chromium has to be sending your Google cookies.
The request to searchdomaincheck is made by the GoogleURLTracker class, which says in its comments:
// This object is responsible for checking the Google URL once per network
// change, and if necessary prompting the user to see if they want to change to
// using it. The current and last prompted values are saved to prefs.
A "network change" will be detected, e.g., by the function
I'm desperately trying not to behave tinfoilhattish here, but this sounds like a cheap way to associate a tracking cookie with an IP address.
According to List of Chromium Command Line Switches, the command-line switch
--google-search-domain-check-url should be able to stop this behavior.
Try invoking Chromium as follows: