Can't they ask claude to create account do some relevant searches and then check if ad is shown?

What makes you think Google has built a profile that shows Claude is a vegetarian?

That's the thing - ask claude to do requests to build such profile

At the scale of a McDonalds ad campaign that seems easy to measure: go to the nearest Target parking lot and ask as many people as you can two questions: "are you vegetarian" and "did you see {description of ad here}". Then compare how which percentage of vegetarians saw the ad compared to the percentage of non-vegetarians saw the ad. There are even companies specialized in market research that will do this in a more representative and statistically rigorous way

Google has no incentive to do so. McDonalds does, but its somewhat costly to scale against ALL the ads they push.

For pay-per-click ads, if the vegetarians won't click, that's a lost opportunity to show a veggieburger ad Google would have made money from.

Google has a lot of incentive to optimize this, and it's a lot of why they (and Meta) spent so much money on AI research that eventually brought us modern AI.

infinite things could confound that analysis

Doesn’t Google earn more from McDonalds if Google provides McDonalds with a higher click through rate? If you’re providing a high click through rate, you’ve probably done the due diligence to screen vegetarians already