I read your post. Several times. Including the title.
Generate project link without owner username
This can be achieved with a simple url shortener.
And if you just want to gift copies of a game, use an external file hoster or some other way to obfuscate the keys/games. I do not grasp why it has to be an anonymous Itch project page. You are already anonymous. I do not know who you are. You can always make a sockpuppet throwaway account to upload a project and be anonymous and change the name of the account later. But I would consider this abuse of the Itch facilities for a mere one time secret santa thing - which by the way is exactly one person to one person. It defeats the purpose to be a public page. Especially the gift part. That sounds like a key to a paid game.
So whatever this is supposed to be, it does not sound like a secret santa. If you have a broker that knows who is who, you can upload the projects on that broker account "anonymously".
Is this maybe some blind jam thing or similar?
Anyways, there are all sorts of problems. It would be much easier to organize the "secret santa" event on some other platform and have the projects on Itch regularly after the event.