I'm not familiar with the American tax system, but couldn't they just not claim that charitable donation when filing their tax returns? Surely it's not illegal

I'm not familiar with the American tax system, but couldn't they just not claim that charitable donation when filing their tax returns? Surely it's not illegal not to tell the IRS about a charitable donation you made, and therefore not get any tax deduction for it, regardless of whether or not you received benefits from the organisation you donated to? (Of course, that wouldn't mean that nobody ever does report this sort of thing to make tax savings, in which cases I would assume you're right it would be tax fraud but again I'm not the person to answer that.)