If $f:\mathbb{R}^k \rightarrow \mathbb{R}$ is nicely behaved, one can view $\delta(f)$ as a distribution (linear functional on $C^{\infty}_c(\mathbb{R}^k)$)- but what if you don't have nicely behaved functions? Is $\delta(xy)$ a valid distribution on $\mathbb{R}^2$? Is it consistent to define $\int f(x,y)\delta(xy)dxdy = \int f(x,0)dx + \int f(0,y)dy$?
My motivation is a slightly more sophisticated example: For spinor-helicity variables $\lambda_i,\tilde{\lambda}_i$, $i = 1,2,3$, which are two-component complex vectors, conservation of momentum is enforced by:
$$\delta^{2 \times 2}(\lambda_1\tilde{\lambda}_1^{T} + \lambda_2\tilde{\lambda}_2^{T} + \lambda_3\tilde{\lambda}_3^{T})$$
(The expression inside is a $2 \times 2$ matrix with complex entries). However there is a similar 'kink' in this set, comparable to the origin in $xy = 0$ (it occurs when all the $\lambda_i$ are parallel and all the $\tilde{\lambda}_i$ are parallel). Is this a well-defined distribution?
The linear functional $$\varphi\mapsto \int \varphi(x,0)dx + \int \varphi(0,y)dy \tag{1}$$ indeed defines a distribution. This distribution can be more concretely described as a measure obtained by adding the linear measures on two perpendicular lines.
However, this does not necessarily mean you want $(1)$ as the definition of $\delta(xy)$. To my understanding, $\delta(f)$ should be the distributional limit of $\delta_n\circ f$ where $\delta_n$ is some approximation to $\delta$, such as $\frac{n}{2}\chi_{[-1/n,1/n]}$. Via the coarea formula, this leads to $\delta(f)$ being the functional $$\varphi\mapsto \int_{\{f=0\}} |\nabla f|^{-1}\varphi$$ which is an integral over surface $\{f=0\}$ with the weight $|\nabla f|^{-1}$.
So, when $f(x,y)=xy$, the above suggests using $|\nabla f|^{-1} = (x^2+y^2)^{-1/2}$ as the weight, obtaining $$\varphi\mapsto \int |x|^{-1}\varphi(x,0)dx + \int |y|^{-1}\varphi(0,y)dy \tag{2}$$ This, as stated, is not a distribution since the integrals diverge when the test function is nonzero at the origin. A regularized version canbe obtained by truncating near the origin, subtracting off the singular part $\varphi(0,0)\cdot 4|\log \epsilon|$, and taking the limit: $$\varphi\mapsto \lim_{\epsilon\to0} \left\{4\phi(0,0)\log {\epsilon} + \int_{|x|>\epsilon} |x|^{-1}\varphi(x,0)dx + \int_{|y|>\epsilon} |y|^{-1}\varphi(0,y)dy \right\}\tag{3}$$