No, the Google Display Network doesn't let advertisers target search queries of users. If you want to access the search intent of users, you have to target the queries themselves with ads on the Google Search Network.
The Google Display Network has a lot of data because it is a large display network with lots of inventory, and it can follow cookies around the web.
I'm ignoring a couple of nuances with this answer, but it is not true that searching for something on Google will lead you to seeing ads for that thing on unrelated websites. If you click on a search result, and end up on someone's webpage, and then they add you to a remarketing list for the Google Display Network -- that's a different story, but that's independent of Google's network, because the same is true whether that website uses AdSense or a competitor.
In other words, we don't share your search queries with anyone. A nuance would be: if you click on an ad, the advertiser knows which keyword triggered their ad to show, so they can guess at the nature of your query that came before the ad click, and then they can intuit your intent. But that's not the same as your search queries, and we definitely don't let people advertise by saying "show my ads across the internet to people who search on Google for ____." So there's never search query sharing between Google Search and the Google Display Network, for a good reason: we want people to trust us when they enter queries, and if you got chased around the internet by ads related to what you were searching for, you might not search as often.
(I'm a PM on Google Search Ads, and I don't represent Google with my answer but rather giving some publicly available knowledge about how the networks work)