$\newcommand\KN{\mathbin{\bigcirc\mspace{-20mu}\wedge\mspace{3mu}}}$If $\omega_{1},\omega_{2}\in \mathcal{D}^{1,1}(M)$ are symmetric, then $\omega_{1}\cdot\omega_{2}$ coincides with the Kulkarni–Numizu product of $\omega_{1}$ and $\omega_{2}$ (maybe up to a sign, depending on the convention). This can be shown by observing directly that both $\omega_{1}\cdot\omega_{2}$ and $\omega_{1}\KN\omega_{2}$ are defined as the double alternation of $\omega_{1}\otimes\omega_{2}$: one first apply alternation on the first two indices, and then on the last two indices.
The "wedge" product for double forms is more general than the Kulkarni–Nomizu product, and makes sense even if the double forms are not symmetric. What Kulkarni showed in his paper is that there is a bundle map of double forms known as the "Bianchi sum" (which he denotes by $\mathfrak{G}$) which satisfies a Leibniz rule with respect to the wedge product. Then, the elements of $\ker\mathfrak{G}$ can be interpreted to satisfy a generalized algebraic Bianchi identity. For $\ker\mathfrak{G}\cap\mathcal{D}^{1,1}(M)$, these are exactly the symmetric tensors, and for $\ker\mathfrak{G}\cap\mathcal{D}^{2,2}(M)$, these are exactly the tensors satisfying the algebraic Bianchi identity of the curvature tensor.
By the way, double forms and this graded structure, along with the generalized Bianchi identity, were introduced even before Kulkarni's paper. See for example Calabi's paper from 1961 (On compact, Riemannian manifolds with constant curvature) or even de Rham's paper (Differentiable manifolds).
I don't know the history of the term for certain, but I guess that the term "Kulkarni–Nomizu product" came up specifically for the product on symmetric tensors since it is the most useful one (e.g. for the decomposition of curvature tensors, which by the way Kulkarni proves for $\ker\mathfrak{G}\cap\mathcal{D}^{k,k}(M)$ in general), and does not require the introduction of double forms in order to be of use.
