I have a dataset of 256 rows with 61 columns/variables. Each row should be considered a vector of dimension 61. If I randomly split it, by rows, in 2 groups, how could I prove that the 2 groups are similar? The origin of the data is biomedical and nonlinear approaches should be preferable.
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.
There are other recent methods like Principal Difference Analyses specifically designed to address these sort of problems. Am not sure if the methd is available as an R package, you can get the concept/algorithm from the manuscript. See http://arxiv.org/abs/1510.08956