PCA and maintaining relationship with target variable
The short answer is that the y_original and x_reduced are still connected to each other, so it is safe to train your data using y_original and x_reduced.  While x_reduced is on a different scale, as you mentioned via eigenvectors, it still is representative of the data that was attached to that observation, just in a different format.  You lose a lot of interpretability as far as what the actual numbers mean which is why it may seem confusing, but it's just a transformed representation of the x_original that (hopefully) contains enough of the x_original variability to make it useful.