That gives you the number but not a comparison. To claim "most searched word on Bing" requires knowledge how much other terms are searched on Bing.

Good point! I guess that would give them the number of people searching for google on bing. Maybe there is enough public information about the totals to say this was the most searched term.

In addition they have Google Analytics on a huge % of pages, so they can track referrals from Bing to those pages, and if the referrals to Google's homepage eclipse those, that's a strong indication.

...assuming Bing isn't preventing that behaviour.

Could you elaborate on this? Is there a mechanism (e.g. HTTP header) which tells Google this when someone types in "Google" in Bing and presses enter?

What if they don't click on the google.com search result?

Yes, it is the referer header, for example this is what my browser sends when clicking a link from Bing after searching for google:

Referer: https://www.bing.com/search?q=google&pq&cvid=3C49B41400954D3...

Google can use that to see your search terms because it in the value of the q attribute.

If they don't go to a Google property then they will still be able to get this information, if the website they do go to uses Google Analytics (which is most of the web).