Assume elements start at index 0. N is odd.
Let mid be an even number close to the middle of the array. Compare a[mid-1] and a[mid]. There is an odd number of elements before mid-1 and an even number after mid.
If both numbers are equal then the unique one is in the left half < mid-1. If they are not equal then the unique one is in the right half ...
This is a classical problem when the array isn't sorted, which has a surprising, almost magical linear solution.
But here, with a sorted array, your idea is great.
The unique element will have an index $u$, and every element $a[i]$ will be exactly the same as either $a[i-1]$ or $a[i+1]$ (which are different).
Assuming that arrays start at $0$, the shape of ...
here, you are multiplying the depth of the "parent" array, instead of the nested array.
multiply it by depth+1
The data structure itself is unambiguously called a contiguous array. Depending on the context and language I would call it "initialized" or "having only non-null elements" or similar to describe the contents.