In regression, standardization is recommended in ordered to assess the relative importance of predictors. However there seems to be an assumption of normality? How would the interpretation work for predictors following different non normal distributions? It seems confusing...
However there seems to be an assumption of normality?
As pointed out by @Arun Jose there's no assumption of normality regarding the independent variables. You can read about the Gauss-Markov Assumptions
How would the interpretation work for predictors following different non normal distributions?
This is a very popular question: check out this answer. We can break it down in two parts: centering and scaling. While the answer to this question concerning scaling is trivial, centering is more interesting.
Centering will result in predictors with mean zero.
[Centering] makes it so the intercept term is interpreted as the expected value of YiYi when the predictor values are set to their means. Otherwise, the intercept is interpreted as the expected value of YiYi when the predictors are set to 0, which may not be a realistic or interpretable situation (e.g. what if the predictors were height and weight?) see here]
Centering plays a role in two scenarios:
The only case I can think of off the top of my head where centering is helpful is before creating power terms. Lets say you have a variable, XX, that ranges from 1 to 2, but you suspect a curvilinear relationship with the response variable, and so you want to create an X2X2 term. If you don't center XX first, your squared term will be highly correlated with XX, which could muddy the estimation of the beta. Centering first addresses this issue [source]
An analogous case that I forgot to mention is creating interaction terms. If an interaction / product term is created from two variables that are not centered on 0, some amount of collinearity will be induced (with the exact amount depending on various factors). Centering first addresses this potential problem same source again and this
Standardization does not change your underlying distribution. It only changes the units of measurement. Also, in regression there is not assumption regarding distribution of your independent variables. The requirement is only that the residuals of your model be normally distributed.