How to give an approximation algorithm for this unusual bin packing problem?
Your problem is known as Multi-Capacity Bin Packing. One of the foundational papers in the area is by Leinberger, Karypis and Kumar, who state a result of Garey, Graham, Johnson and Yao that in the case of $d$ constraints (in your case, $d = n$) many natural algorithms give a $d+1$-approximation.