Is it possible to determine if 2 arrays contain the same elements (ignoring duplicates) in faster than O(n log n) time?
I believe a hashing approach should give you $O(N)$ time, where $N$ is the length of the longer array (we can relax the constraint that they are the same size).

Simply put all the elements in the first array into a hashset, and then check whether each element in the second array exists in the hashset.

You could even account for duplicates by instead using a hashmap and storing the counts of each element in the first array. When checking elements in the second array for inclusion (or nonzero value), decrement the value of each key that is a match.