According to Google [1], they keep client IP addresses and DoH headers for up to 48 hours and then strip them after that, and don’t use the logs for any personalization.
Everyone else here is assuming you can’t trust what they say, but it’s worth pointing out that, if they’re telling the truth, none of the rest of this privacy discussion matters.
Sure, which is why Oblivious DNS would be a nice upgrade. One less thing to worry about.
But I expect that, compared to 8.8.8.8 or 1.1.1.1, the user experience will be unaffected, and there will be no measurable improvement in user privacy. (Because we have no way to measure it unless something bad happens and it can somehow be traced back to one of these DNS services.)
Everyone else here is assuming you can’t trust what they say, but it’s worth pointing out that, if they’re telling the truth, none of the rest of this privacy discussion matters.
[1] https://developers.google.com/speed/public-dns/privacy