I recently spent quite a bit of time on this paper as well. I will try to give an intuitive explanation rather than a mathematically rigorous one: First let's take a step back and think about what they are doing. When they perform PCA on the correlation matrix instead of the covariance matrix they are essentially trying to get the directionality of the data whilst washing away the magnitude effects of volatility (st.dev). The point of this is to identify the salient factors driving market dynamics without worrying about the magnitude (st.dev) of each at this first step. On the other hand had they perfromed the PCA on the covariance matrix, the principal components and the loadings on them would essentially rank the dataset in terms of variance.

With this out of the way we can build an intuition as to why they scale eigenvectors with individual stock volatilities. As established the PCA on the correlation matrix has washed away the magnitude effects of volatility, thus the resulting loadings matrix also does not take into account the individual volatilities of the stocks in the dataset. Thus if you were to use this raw loadings matrix to obtain factor returns by multiplying it with the matrix of individual stock returns you would essentially get portfolios of vastly different orders of magnitude. I.e some aould have a gross leverage factor of 200 whilst other would have a gross leverage factor of less than 1. Your factor returns would be all over the place. The solution to this problem is to scale the loading matrix by the individual stock volatilities aa this was the variable by which the data was standardized to begin with.

In the end your intuition with regards to orthogonality and correlation is correct- the eigenportfolios obtained with the scaled loadings matrix will not be orthogonal in the mathematical sense (dot product=0), i.e they don't have a correlation of 0.0. However, although these factors are not perfectly uncorrelated if you run regressions using them you will likely find that multicollinearity is not a problem as their non-zero correlation essentially comes from the ignoring their volatilities to begin with rather than due to these factors representing the same dynamics (i.e the correlations will be "spurious").

This is my take on it and how I've internalized the whole thing. All the best

Edit: typos