Incognito windows don't block cookies, they just sandbox them from the rest of your cookies and are deleted when you close the Incognito windows, so the behavior you describe isn't too surprising.

But what does that have to do with Chrome's search/history? If I use Chrome's search bar to search using StartPage there's no way for Google to know my searches or even the fact that I'm using StartPage.

I'm just saying you can get the same behavior as DDG's shortcuts with Chrome without using Google at all.