I do believe they mean an iron nail, and they are not denying that water dissolves nails, they are just saying that Coca-Cola does much faster.
I think it is finger nail, because the sentence continues with osteoporosis - a small piece of finger nail could possibly dissolve relatively quickly if you can keep the liquid acid after opening the bottle, but I still believe it will take more than days. I heard the story of an iron nail dissolving in Coke a few decades ago at school when a teacher was trying to convince us that Coke is evil - I just realised it might have originated from a translating mistake since nail and nail are two different words in Finnish.
Can you find pH paper, Brannigan? If you'll try to dissolve stuff, it would be useful to know the acidity of each liquid, even water can be slightly acid or slightly alkaline