How can I measure if a population has the same distribution as other? Is there a population similarity index of some kind which could help me determine if two populations in two different datasets are the same or at least similar?

The datasets have the exact same number of variables.

I would like a measure which assess the similarity variable-by-variable, but an overall measure works too.

The context of my question is that I would like to know if a variable is suitable or not for using in a classification method given the similarity of it between populations.
What I'm gonna say might seem too simple, but I assume, it might not be bad to fit a multivariate distribution (like Gaussian) to each distribution and then figure out what the mean and covariance matrix are. Mean might not depict so much information but the variance and correlation that are exposed in the covariance matrix might be helpful. 

For the case of determining if a variable is useful or not, analyzing its correlation with other parameters might turn out to be useful. 

(These are my own thoughts)