Not yet an answer but you can see $f$ as an element of the symmetric power $S^d(V^{\vee})$ for $V$ a vector space of dimension $n+1$. You can take $V=S^n(W)$ with $W$ of dimension 2. That should give something like your representation for $g$. The indices labeling the $x$ basis of $V^{\vee}$ become labels for monomials of degree $n$ in your $y$ variables.

More explicitly, if
$$
f(x_0,\ldots,x_n)=\sum_{i_1,\ldots i_d=0}^n f_{i_1,\ldots,i_d}x_{1_1}\ldots
x_{i_d}
$$
with the (physicsy) tensor $f_{i_1,\ldots,i_d}$ being symmetric, then
the associated form is
$$
g(y^1,\ldots,y^d)=\sum_{i_1,\ldots i_d=0}^n f_{i_1,\ldots,i_d}
(y_0^1)^{n-i_1}(y_1^1)^{i_1}\ldots
(y_0^d)^{n-i_d}(y_1^d)^{i_d}\ .
$$
It is of course essential to now see $f$ as a symmetric multilinear form rather than a homogeneous polynomial, i.e., to work with $n$ distinct pairs of binary variables $y$. This is the idea behind the **symbolic method of classical invariant theory** (see my answer to this MO question for more details).