Hi, I'm not quite sure I understand you.
You ask something, that yourself answer with the link, but then it seems that you are really looking for a DRM system.
First, directly answering your question, in the same link you share you have the following API:
https://itch.io/api/1/KEY/game/GAME_ID/download_keys
You can use this API call to verify that someone has a valid download key to download the game
In other words, with that you can know if a certain user owns your game or not. However, it seems that this is not what you are looking for.
From what you describe, it seems that you are more interested in knowing that indeed the person who is using the PC is a valid itch user and that they have bought your game. In that case read about OAuth which seems to be what you are looking for.
https://itch.io/docs/api/oauth
Regardless of that, since you are talking about piracy, I think what you are really looking for is to incorporate a DRM system, and Itch does not incorporate any such tool, if you want to implement any DRM mechanism in your game, you must implement it yourself.