I am a native English speaker and I don't get it either. I googled 'what are teeth and nails made of' to try and give myself a clue and got this delightful generative bullshit response:

"Fighting tooth and nail" is a phrase meaning you're in a desperate struggle, using everything available to you. By capitalizing it, it also turns tooth and nail into people with non-gendered names. That's all there is to it, it's just a pun.