This might be a controversial take, but at the heart of it, yes, I think it is a bit of a fancy yet powerful data structure (it inherently uses data to create an 'algorithm' unlike most conventional algorithms). The "training" is all about using data to set the weight and biases across the neural network and hence create this 'algorithm'. This essentially means that the weights and biases are in fact your data, massaged and extruded to a humanly unrecognisable level.