You can also claim it on Itch to download later, like "add to library".
Free Video Game Giveaways
This is a place to post limited-time giveaways of full video games that are then yours to keep forever at no cost.
See the sticky post for full details of what to post, how to post it, and other useful resources.
Keep it civil and keep it chill!
Yep, that's the case with almost all itch.io games I've grabbed from here, weirdly a few of them don't offer the option though
I think that a game has to be "purchaseable" for $0 to have a "claim" option. If it's just "free", it won't have a "claim" option. Some authors seem to switch this setting on purpose to stop people from keeping free download access to their game after the free period has ended. Sure, we could just keep the files backed up ourselves, but a) how many people really do that, and b) the author could release updates and DLC later to encourage a purchase.
Makes sense!