In addition, Chrome is distributed by Google itself and they clearly advertise it as made by them, better and safe. They make it explicit that it is entirely under their control. So, it would be logic for a standard non tech user to expect that google do not track them when they are using the "google" chrome incognito mode.
Keep in mind that a lot of users would not know the difference between a client and a server.
Also, remember that google do everything to blur the lines when it was convenient for them:
When you are logged in Google, your own browser is logged with a special profile icon. If you look at it you can launch a Google search directly from the url bar or from the "new tab" page.
Even as a power user, can you easily reply to the following question? is the new tab your browser domain or Google domain? Hint: some content here is dynamic depending on your Google account...