A typical household connects to a single router when accessing the Web. To the external world they will all share the same public IP address.
If I understand correctly, individuals are often tracked by their IP address, besides other identifiers. Which may be one possible cause.
How can I preserve the privacy of each family member and avoid the annoyance of ads and content being misattributed to other family members?
You could use a search engine like DuckDuckGo that doesn't remember your search history.
You could look into using a network-wide ad blocker such as Pi-hole: https://pi-hole.net (requires some flavor of Linux).
Since it is probably the IP that is tracked, maybe implement a VPN service for the whole family or have your router to scheduled dial-ins to reset your dynamic IP (if you get one).
For web browsing I would recommend an ad blocker, It think it is pretty much mandatory to enjoy the web. Also hits sites with responsible advertising, you would have to whitelist them manually as they are often dependent on it.
The other general privacy suggestions mentioned like uBlock Origin, Pi-hole / similar, DuckDuckGo, etc are all good as well.