If you got the itch download, it should give you the option to link the game to your account. Make sure you are signed into your itch account when you click the link.
Hm that doesn't seem to be working. The link takes me to a page where I can download. But there is no option to link that I can see present. There is an option to add to collection. But that just creates a link to the game page in my collection. If I follow the link it still wants me to purchase the game & does not show it as owned.