How do GANs learn category distributions

I'm currently getting more into the topic of GANs and Generating Models. I've understood how the Generator and Discriminator work together in optimization to generate synthetic samples.

Now I'm wondering how the model learns to reflect the occurance frequencies of the true entries in case of categorical data. As an example, lets say we have two columns of entries (1,A), (1, A), (2, A), (2, B) The model, when trained would not only try to output real combinations, so e.g. (1, A) and not (1, B), but also it would generate (1, A) more often than (2, .) due to its more frequent occurance. Is this being trained when providing the training examples to the model?

2. Question: How would the training look like for 1D categorical data: (A, B, A, B, C).

Here clearly, the discriminator would never output fake, as all generated examples in 1D are real examples. So I assume a GAN would then be able to learn the distribution of the categories and when being run after training, generate more 'A's than B and C. Is my though process correct?

Thank you!