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Consider "Volume" as a function from set of $d$-dimensional convex bodies to real numbers. This function is homogeneous of degree $d$ (under rescalings of the convex bodies). Minkowski's theorem asserts that this function is in fact the restriction of a symmetric multilinear (with respect to Minkowski addition) function on $d$-tuples of convex bodies called "mixed volume".

This is akin to the relation between a quadratic form and corresponding bilinear form, with polarization identity (generalized to $d$-linear case) as usual implying that if such mixed volume function exists it is unique (see https://mathoverflow.net/questions/71952/do-the-elementary-properties-of-mixed-volume-characterize-it-uniquely/71980#71980 for example.)

Of course an arbitrary homogeneous function of degree $d$ on a vector space is not a homogeneous polynomial, and so more likely than not it does not come from restricting a multilinear symmetric form.

Why does volume? I other words, why should mixed volumes exist?

Of course it's a theorem they do, but the proofs I've seen proceed by induction on dimension and I have not been able to extract any intuition from them.

After some literature search, I found two things:

1) This "polynomial" perspective is well-developed in Hörmander's book "Notions of Convexity". In the Appendix A Hörmander discusses what it means for a function on (an open convex cone in) a vector space to be a polynomial (in a way which for finite dimensional vector spaces is equivalent to being polynomial in some, hence every, basis) -- his definition is that the function becomes a univariate polynomial when restricted to any line (not necessarily through the origin). He discusses how such functions are obtained from multilinear forms by polarization, and proves Proposition A.1 which in context of convex geometry establishes equivalence of concavity of $Vol^1/d$, the Brunn-Minkowski inequality, and the isoperimetric inequality, among other things.

In the main text (p. 75) he embeds (by considering support functions) the cone of compact convex sets in $mathbbR^d$ into the vector space of continuous function functions on the dual $(mathbbR^d)^*$, thus producing a vector space in which one can state this polynomiality (this is a nice concrete version of taking the Grothendieck group of the monoid of compact convex sets under Minkowski sum).

More importantly for this question, Hörmander's proof of Minkowski's theorem, while still inductive on dimension, made it clearer (to me) why volume is polynomial. After reducing to the case of polyhedra, we are reduced to the following observation: the volume of a polyhedron is a sum of volumes of pyramids from an interior point to its faces (or in fact from any point, but we would then need to take oriented volumes). The pyramids participating in this sum for the polytope $lambda_1 K_1 +lambda_2 K_2$ have the height $lambda_1 h_1 +lambda_2 h_2$ which is (degree 1) polynomial in $(lambda_1, lambda_2)$ and the "area of the base" is a degree $d-1$ polynomial in $(lambda_1, lambda_2)$ by inductive hypothesis. Thus each "pyramid volume" is a degree $d$ polynomial, and hence so is the volume itself.

I think if one follows through the induction one should get a kind of "orthogonal barycentric subdivision" of $lambda_1 K_1 +lambda_2 K_2$ with each simplex constructed from the simplexes and altitudes of $lambda_1 K_1$ and $lambda_2 K_2$ in the way that makes their volumes a product of $d$ linear polynomials in $(lambda_1, lambda_2)$.

2) The combinatorics describing the details of the "following through the induction" above should be essentially described by what is called "mixed subdivision" of the Minkowski sum, which also explains why volume of convex sets is polynomial. Roughly, one takes the join of $lambda_1 K_1$ and $lambda_2 K_2$, embedded into $mathbbR^d+1$ (in this context this is called "Cayley embedding"), triangulates that and projects to the original space. One obtains a triangulation of the Minkowski sum via "mixed simplexes"(that is projections of joins of simplexes in triangulations of $lambda_1 K_1$ and $lambda_2 K_2$), each of which clearly has volume which is polynomial in $(lambda_1, lambda_2)$; thus the whole Minkowski sum has volume which is polynomial.

This is described in detail in chapter 9 of "Trinagulations" book of De Loera, Rambau, and Santos.