privacy – How can Instagram make Chrome try to sign in and display my username when I use my incognito mode?

Instagram can not and does not. Chrome is. Chrome knows your Google Account because you are signed in and that Instagram allows you to sign in with your Google Account. So when you visit the Instagram website, they offer to log in automatically for you. I do not know why this would trigger in Incognito mode; it seems like a possible bug.

This should be disabled via the Settings menu – look under the Privacy & Security submenu. Chrome is not available to find the exact path.