No, it's really difficult for 'them' to find out whether you actually purchased something. All the tracking stuff is on social media and the product browsing pages. But not on checkout. Shopping sites can't allow all that stuff there because of security and privacy issues. So big tech doesn't even know whether you bought something. They only know what you have seen. That's why they keep showing it.

Conversion tracking is not merely a thing, I'm under the extremely strong impression that it's standard.

The owner of the shopping site can do it. They know which campaign lead to a sell.

But I have never heard of them passing it on to whatever 3rd party trackers they have on their sites. THAT would be a privacy violation.