It does the opposite - the PCA graph shown is 2D, meaning points in the N-dimensional space are projected onto a single plane. In this projection, information is lost - i.e. point-clouds that are distinct in 3D may overlap in 2D. While we are shown the axes along which the data is most distinct, the true clusters are even more distinct, as every added dimension would contribute additional 'distinctness'.
(No, of course they shouldn’t. It’s just that extracting features to artificially cluster points along the most characteristic axes is what PCA is for.)
[0]: https://commons.wikimedia.org/wiki/File:3D_PCA_plot_of_Xavan...
I believe I described PCA accurately, so could you elaborate which part of my statement is wrong?
> extracting features to artificially cluster points along the most characteristic axes is what PCA is for
How is the clustering "artificial"? Because if you generate data without clusters (e.g. points evenly distributed within a sphere), applying PCA to it won't show clusters either.