What is the complexity class of this variant of Subset sum?
This is NP-hard. The associated decision problem is NP-complete.

There are various ways to prove that.  For instance, there's a straightforward reduction from exact cover; let the target array be all-ones, and then you have an instance of the exact cover problem, which is NP-hard. 

Your problem is an instance of multi-dimensional subset sum (or multi-dimensional knapsack), for which you can find algorithms and approximation algorithms in the literature.

Related: https://cstheory.stackexchange.com/q/19976/5038, How to find subset of vectors whose sum has certain characteristics, https://cstheory.stackexchange.com/q/21865/5038, https://en.wikipedia.org/wiki/List_of_knapsack_problems#Multiple_constraints