Is there a way to add free itch.io game assets to your account, or must I download them to keep them

When Oxenfree got removed from itch the option to download it got removed as well. Based on that experience you better download your materials, though tbf I don't know how universal that is.

I believe you can just add it to your collection in addition to the pay zero dollars or what you want to pay thing.

Sorry, didn’t really answer the message in your comment, but as DeliberateTaco says, for each item, you have to click the page and add to collection. No easy or quick way to do it unfortunately.