That’s a lot of words to say that, if you encode a class of things as numbers, there’s a formula somewhere that can approximate an instance of that class. It works for linear regression and works as well for neural network. The key thing here is approximation.
I can construct a Gaussian process model (essentially fancy linear regression) that will fit _all_ of my medical image data _exactly_, but it will perform like absolute rubbish for determining tumor presence compared to if I trained a convolutional neural network on the same data and problem _and_ perfectly fit the data.
I could even train a fully connected network on the same data and problem, get any degree of fit you like, and it would still be rubbish.