Measurement errors in CFA standardising latent factors

I came across this magnificent post from Rose Hartman (http://www.understandingdata.net/2017/03/22/cfa-in-lavaan/#writeup) that suggests standardising latent factors (constraining them to have a mean of 0 and a variance of 1) in order to allow free estimation of all factor loadings when performing CFA. I did it with the lavaan package in R by setting std.lv=TRUE when calling cfa().

My question is: am I standardising unique factors (measurement errors) too, by doing so?

No, the latent variable standardization protocol with CFA only standardizes the latent variables. If, before you ran the analysis, you standardized your manifest measures, then the factor loadings can be interpreted exactly as correlations (between latent factor and measured item). However, even then, your error variances will be constrained by the factor loading (correlation). In particular, if the manifest item has $\lambda$ factor loading, than the variance of the standardized manifest item with be estimated as $1-\lambda^2$. Thus, the measurement errors will only be standardized if all $\lambda$ are zero...which defeats the purpose of CFA (i.e., dimension reduction), as it suggests each item is its own “dimension”.