The problem is that I'm using dimensional in two different senses. A vector with N elements, for example, is a one-dimensional array of numbers operating on N dimensional spaces, and so on for matrices and tensors.

If you're doing graphics programming you're operating in three and four dimensional spaces mostly (four dimensional being projective spaces), because that's the space you're trying to render in two dimensions. But you'll rarely need anything higher than a matrix (a two-dimensional data structure) for operations on that space.

If you're doing physics you're operating in three, four, and infinite dimensional spaces, mostly. And you'll routinely use higher data structures -- even things like moment of inertia for rigid bodies can't really be described without rank 3 tensors (a three-dimensional data structure).

In statistics and machine learning, you're operating in very high dimensional spaces, and will find yourself using non-square tensors especially (in the other areas everything will be square). The data structures will generally be high dimensional as well, but usually just a function of model complexity; so maybe 4 or 5 dimensional data structures.