What are x variable and y variable in word2vec model if it is supervised learning
First I would call it self-supervised in the sense that you dont need human labeling, but still need labels for NN to learn.

Both CBOW and skip-gram are "combined" into word2vec but they are shallow-neural-network architectures (albeit different ones).

Since they are different architectures, they do different things:

CBOW learn to predict the word by the context. Or maximize the probability of the target word by looking at the context.

skip-gram model is designed to predict the context so given the word xyz it must understand it and tell us that there is a huge probability that the context abc__dfg is what "surrounds" xyz.

Given that, you could try to deduce yourself what should input and output data look like to model this desired input output of CBOW, skip-gram (matrix representation of words and than probabilities and a vector representing a word and multiclass output).

Check also this answer