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.
Yes google stores your activities on google browser you do every day and they will try to display ads based on your interest so they can get more clicks for them and more profit also.