Clarification on Earmarking Donations

If I cannot earmark my donation to a specific person that I want to benefit from my donation, how is it possible that at Christmas Time when the newspaper shows that photos of kids and what they want for Christmas, that I can give to that organization to buy the gift for that child and specify that it is for the particular child?

Similarly, how does a donation to Make A Wish become tax deductible when I am picking the child and contributing to his/her specific wish?