Of course there is, but if you perform an operation on a multidimensional array, there is no guarantee it corresponds to an operation on tensors, ie. the resulting tensor may depend on the basis.
Sure, if you perform an arbitrary operation on a multidimensional array. But the same is true of any representation of any mathematical object. It makes no physical sense to take the sine of a mass, or two to the power of a length. But that doesn't mean that whenever someone says "oh, the mass of an object is a real number" I need to nitpick them.