How is this information logged/preserved, and for how long? Just because it isn’t being used for ads doesn’t mean a person is comfortable filling up a database with their activity.

IMO, ads are probably the least worrisome way the data could be used. A boring but scary example is that aol search history leak (which is still searchable today):

https://searchids.com/

This person is identified by name for example: https://searchids.com/user/19431784-joann_whitman