You are aware that Itch is a store front for people to publish and sell their games? Like, publicly.
It is not supposed to be a file exchange from one person to a single other person.
Just use one of the regular file hosters for that. They do not have this "uploader" information and are rather anonymous from perspective of the downloader.
Also, game urls on Itch are always connected to the account and profile that published the project.
Oh, and if this is about urls only and not about files and keys and such, just use an url shortener. At the latest when viewing the page, the gifted person would know who made the game. Or when starting the game.