For clustering and other techniques for mixed data (numerical and categorical), Gower's distance is usually more preferred than Euclidean distance because the former computes distance differently for numerical data and categorical data. For the numerical data, Gower's distance takes the normalized difference into account. For categorical data, it only registers if the category is the same or not (end result is 0 or 1).
My question is: if we use Euclidean distance on the data after standardizing the numerical features (i.e., normalizing each column with MinMax or StandardScaler) and one-hot encoding of the categorical features, is it equivalent to the Gower distance? Well, at least qualitatively, up to the standardization procedure?
Or I am missing something? Perhaps, it would make difference when categorical data has more than 2 categories and hence only detecting difference in 0 or 1 is not enough?