Browsers have access to all of your data and they don't need to guess based on cookies and whatnot if searches are from the same person. So naturally, a browser is the perfect place to mine personal data for advertising purposes. If the browser is open source, the treatment of ads can be audited to ensure no personal data is being leaked.

For example, if you frequently visit gaming related websites, then the browser will know to show you gaming related ads. Google would only be able to do that if you use their search engine or if enough of those sites opt in to sharing data with Google (e.g. amp links, Google Ad integration, etc). So Google's ads (or any other kind, for that matter) are by default less relevant because they have less information than something served by a browser.

The difference between doing it browser side vs server side is where privacy comes in. With server side (e.g. Google's method), your data is sent to Google and they can then do whatever they please with it (depending on jurisdiction and what laws apply). With browser side, your data stays on your machine, so it never needs to go to the browser vendor or advertisers, so it cannot get sold or used for anything outside of the browser. The only thing advertisers and browser vendors would know is how many times an ad was shown and how many times it was clicked on, and that could not be traced to you specifically unless you do something to opt in to that. That's it. No privacy violation.

So since browsers have access to more data than an advertising company would, they can be a lot more relevant. Browser vendors could pay users a bit to allow anonymous usage statistics to refine their model, but I don't think that's necessary.