Is learning the moves a special case or just the same sort of thing that happens as the AI learns strategy? If you take two different neural networks and teach them each how the pieces move, what checkmate is, etc. will the two networks look identical or is there a random element that means that two networks with the exact same number of nodes, running on the same processors, etc. will not look identical even though they both know exactly the same things?
How does learning the moves of chess show up in a neural network?
If you have 2 neural networks that have been initialised in the same way, are trained with the exact same sequence of samples, and there is no other source of randomness, then they will be identical at the end of training.
So, the answer to your question is: it depends on the learning algorithm, how you initialise the neural networks, and how you sample the data and feed it to the neural network. There's no magic!
Finally, what they learn is encoded in their weights and how these weights are used to produce the output (i.e. the architecture). It's not easy to explain intuitively what the neural networks are exactly doing to produce the output (so they are sometimes called black box models), but there are explainable AI techniques that can be used for this purpose.