What is the additional structure present in homogeneous polynomials when used in representation theory? Representation theory of $SU(2)$ uses homogeneous polynomials in two variables of degree $n$ for the vector spaces which are used to represent the actions of group elements.
I am trying to understand the reason for this. Since the homogeneous polynomials form complex vector spaces themselves, they are isomorphic to some ${\mathbb C}^n$. So why not use ${\mathbb C}^n$ for this purpose? Obviously there is some additional property which comes in handy. I am interested in knowing what this property is, because the use of these homogeneous polynomials, for me, comes way too much as unobvious surprise.

You're certainly correct that any finite-dimensional complex vector space is isomorphic (as complex vector space) to some $\mathbb C^n$... so there's no necessity of modeling those repns in terms of polynomials.
However, there is a small advantage, namely, that spaces of homogeneous polynomials in two variables do model "symmetric powers" of a two-dimensional vector space, in a fashion that is compatible with the group action... and without having to talk about "symmetric powers" or "tensor products" or other multi-linear algebra before starting. That is, we can look at those irreducibles right away... rather than post-poning it till after more apparatus is set up.
That is, the action $SU(2)$ induces on the homogeneous degree $n$ polynomials from the matrix action on the "indeterminates" $x,y$ is
the symmetric-power action/representation... without having to define it otherwise. :)
