There is a high competition with google's rival - Facebook.
Google knows that what people are searching but doesn't know who?
Facebook know exactly who searches like location age preferences and lots of other information, that's why google is attempting to be in every part of our lives.
No, I think you aren't completely right, when a user uses the google browser, he is often directed to signing up for a google account or logging in, and google also remembers accounts from IP addresses and saves personal information too. Facebook on the other hand is a competition when it comes to advertising, but both of them support each other indirectly due to user distribution.
If not completely then mostly i am right i think,
Facebook knows in detail what one of their user searches, their age, updated photos, location all information.
Google has less information than facebook even though is trying to get more and failed with it social Google+