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.

I found out that itch.io is missing app authorization to find out if users own the games actually. What I want is to find out if players own my game. And if so, they can play the game. itch.io is missing that thing, so it's easy to get pirated for our itch.io games.

As in contrast to which games? If you can get pirated copies of AAA games, what do you think your games are?! 

DRM is a problematic concept in many ways. Foremost, that it only punishes legit players. You will not sell more games, because a game has drm. The creator might have the fuzzy warm feeling that the game is harder to copy. But the concrete cold reality is, that  no one will buy the game because it has drm, but many will intentionally not buy it, if it has one. Just read some steam reviews of games that implement additional drm.

For any kind of online experience, drm is moot anways.   You have your servers and login credentials, a thing like client side drm does add nothing.

The api you mentioned requires you to have a server. Wich brings us back to the online experience. And there are users that will be very, very suspisious, if an offline game requires internet connection. And quite frankly, I would not trust a small time indie dev to handly any such thing at all.

But from what I understood, you could implement some login type drm   with this oauth thingie. But after having a look at your creators profile and games, your concerns about pirated versions of your games are purely academical. And to be more specific, I for one would not buy any of your games, if I had to login onto your servers to play  them. For the single reason, that I do not trust a small dev like you to still have that login server in two years.