I believe (I haven't read the study in detail) that their results show that different people get different results when not logged in. They didn't actually correlate this with anything like purchase history or individual user info. Others ha e hypothesized that the differences are due to location.

In other words, the study proves that different people get different results. It doesn't prove, as they claim in some press, that this is because Google tracks you while you aren't logged in. It's misleading on DDGs part to claim such of that's the case.

(I work at Google, but have nothing to do with ads).

As a broader point, just because you have a null hypithesis does not mean the study is biased. That's completely not how academia works.

Bias as the gp used it doesn't mean statistical bias which also isn't what you mean, but more like "even if these results are technically accurate, they may be misleading", a common example would be p-hacked results. This, contrary to what you say, often involves inventing a hypothesis after the fact, such that it is easily disproven with your data.

You make a good point that the fact that there are different results doesn't mean that they are personalized. It could simply be that Google is providing different search results so they can get better data on what people want to click. This would be a benign — indeed legitimate — explanation for the DDG study results.

However, there isn't evidence that this was what was happening, and Google hasn't exactly covered itself in glory on the privacy front so I tend not to give them the benefit of the doubt on this front. I'd love to see a response from them addressing some of the points from the DDG study.