If it is stored entirely on the machine you own, then you are processing and storing the data.
Next, Google will be downloading a database of 2 billion ads to your PC to check against all your private data and decide which ad to show you for anything you search in the future.
And because its all on your PC, no GDPR issues - you are processing your own data.
If you have full sovereignty over your data and your computer and you decide to use software that downloads ads and blasts them to you, at some point even in the best jurisdiction you're responsible for what you install