In my mind, the red cross symbol has become a universal symbol for 'medical service or supplies', and not just for this one organization.
Perhaps they should change their logo. Or just be content that they 'own' one of the most recognizable symbols in history.
What other symbol would you use for medicine/medical? Just look at any first-aid box, and it has a red cross. Even nurse dress-up costumes.
Of course if I tried to start an NGO using the symbol, I would rightly be sued. But using it generically, in games for med-paks or whatever, etc, seems fine to me.