It may not be the only reason, but you’re being too generous if you don’t think this was at least one of the reasons they did it.

Other than some abstract “branding” campaign, I cannot really see many other reasons why they would be doing this.

And as someone who worked in adtech in the past, it was very well known that Google used their domain as their tracking cookie domain as it’s nearly impossible for adblockers to just block without crippling other functionality. So they even have a history of using precisely these types of techniques.