They could also just have an extra option for people who like to donate in a fully private manner without the need to insert fake info.
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They probably could do such option for small donations, but that could make donation form too complex. Also it could be risky for charities, if they're required by law to collect personal info for donation above certain amount, then if someone used 'anonymous small donation' option but made a large donation, it would create a lot of mess.
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Doing effort to claim back a certain percentage of your donation is nonsense. Donate in full, or don't.
Not sure what you mean by that. If you meant claiming the tax back - it has nothing to do with claiming % of donations back. You claim back the tax that you've already paid on your income (as charity donations are tax-free).
There's also thing called gift-aid (in UK), which allows you to donate the tax (you could claim back) to charity. So effectively for £1 donated, charity would get £1.25. But again, the charity would need your details + 'gift aid declaration' for that.