I'm pretty sure that a multidimensional array was the original implementation of tensors and the more fancy linear forms formalism came later in the twentieth century. Then the question was how do these multidimensional arrays operate on vectors and how do they transport around a manifold. (Source: first book I read on general relativity in the 1970s had a lot of pages about multidimensional arrays and also this good summary of the history: https://math.stackexchange.com/questions/2030558/what-is-the...
(Sort of like how vectors kind of got going via a list of numbers and then they found the right axioms for vector spaces and then linear algebra shifted from a lot of computation to a sort of spare and elegant set of theorems on linearity).
> The transformation law for a tensor behaves as a functor on the category of admissible coordinate systems, under general linear transformations (or, other transformations within some class, such as local diffeomorphisms.) This makes a tensor a special case of a geometrical object, in the technical sense that it is a function of the coordinate system transforming functorially under coordinate changes.[24] Examples of objects obeying more general kinds of transformation laws are jets and, more generally still, natural bundles.[25][26]
Mathematicians are comfortable with the idea that a matrix can be over any set, in other words matrix is a mapping from the cartesian product of two intervals [0,N_i) to a given set S. Of course to define matrix multiplication you need at least a ring, but a matrix doesn’t have to define such an operation, and there are many alternative matrix products, for example you can define the Kronecker product with just a matrix over a group. Or no product at all for example a matrix over the set {Red,Yellow,Blue}.
Tensors require some algebraic structure, usually a vector space.
(Sort of like how vectors kind of got going via a list of numbers and then they found the right axioms for vector spaces and then linear algebra shifted from a lot of computation to a sort of spare and elegant set of theorems on linearity).