This question is a follow-up to a question about the theory of polynomials.
It should be quite clear by now that matrix theory and linear algebra are quite different topics. As the various answers to that question clearly state, the difference is largely because of basis choice. But we know how to do a lot of abstract linear algebra, without ever having to choose a basis.
But when one switches to polynomials, the situation changes: even very general definitions of polynomials (like that in Lang's Algebra, for example) still use monomials in the standard basis. Why that basis, and not the rising factorial basis, or falling factorial, or Chebyshev basis, or any other? And the most important question: why, in fact, choose a basis at all?
Which leads to the question in the title: Are plethories a theory of basis-free polynomials?
I think the motivation should be clear: if going 'basis free' was so successful for linear algebra, shouldn't we also expect similar success with polynomials? Unfortunately, I have not been able to find any work which I could readily understand as being about developing a theory of basis-free polynomials.
EDIT: After reading the current comments and answers, I am starting to think that perhaps what I am really seeking is a theory for basis-independent polynomials, or even basis-generic rather than leaping right away to basis-free. There are many applications of polynomials where bases other than the monomial one greatly simplify both reasoning and computations (which is what I am eventually after), but the tools for doing this seem to be hard to find.