Group similarity by high dimension vector comparison
You can't actually prove that the two groups are similar but you can establish a confidence level/threshold. Furthermore, it is possible that the two groups won't be similar (depending on your threshold for similarity) if, for example, only one of the two groups contains strong outliers.

That said, you can make comparisons based on assumptions regarding the underlying data distribution. For example, if the data can be assumed to be distributed as a multivariate normal distribution, you can use Hotelling's two-sample T-squared statistic (a multivariate generalization of the Student's t-test) to test your confidence interval.