The case you describe is the general case since every measures $\mu$ and $\nu$ are absolutely continuous with respect to $\mu+\nu$. More precisely, there exists $h_{\mu,\nu}$ with $0\leqslant h_{\mu,\nu}\leqslant1$ everywhere such that $\mu=h_{\mu,\nu}(\mu+\nu)$ and $\nu=(1-h_{\mu,\nu})(\mu+\nu)$. Thus one can define an intrinsic product $\mu\odot\nu$ by $ \mu\odot\nu=h_{\mu,\nu}(1-h_{\mu,\nu})(\mu+\nu). $ When $\mu$ and $\nu$ are absolutely continuous with respect to the Lebesgue measure (or any other measure of reference) with densities $f$ and $g$ respectively, then $\mu\odot\nu$ is absolutely continuous with respect to the Lebesgue measure with density $f\odot g$ defined as follows: on $[f+g=0]$, $f\odot g=0$, and, on $[f+g\ne0]$, $ f\odot g=\frac{fg}{f+g}. $ This product $\odot$ on measures is commutative (good), associative (good?), the total mass of $\mu\odot\nu$ is at most $\frac14$ times the sum of the masses of $\mu$ and $\nu$, in particular the product of two probability measures is not a probability measure (not good?), $\mu\odot\mu=\frac12\mu$ for every $\mu$, and finally $\mu\odot\nu=0$ if and only $\mu$ and $\nu$ are mutually singular (good?) since $\mu\odot\nu$ is always absolutely continuous with respect to both $\mu$ and $\nu$.
Edit To normalize things, another idea is to consider $\mu\Diamond\nu=2(\mu\odot\nu)$. In terms of densities, this corresponds to a harmonic mean, since $\mu\Diamond\nu$ has density $f\Diamond g$, where $ \frac1{f\Diamond g}=\frac1{2(f\odot g)}=\frac12\left(\frac1f+\frac1g\right). $ In particular, this new intrinsic product $\Diamond$ is idempotent (good?), commutative (good), and not associative (not good?).
Edit A canonical product concerns probability measures and transition kernels. That is, one is given a measured space $(X,\mathcal X,\mu)$, a measurable space $(Y,\mathcal Y)$ and a function $\pi:X\times\mathcal Y\to[0,1]$ such that, for every $x$ in $X$, $\pi(x,\ )$ is a probability measure on $(Y,\mathcal Y)$. Then, under some regularity conditions, the product $\mu\times\pi$ is the unique measure on $(X\times Y,\mathcal X\otimes\mathcal Y)$ such that, for every $A$ in $\mathcal X$ and $B$ in $\mathcal Y$, $ (\mu\times \pi)(A\times B)=\int_A\mu(\mathrm dx)\pi(x,B). $ In particular, $B\mapsto(\mu\times\pi)(X\times B)$ is a probability measure on $(Y,\mathcal Y)$.
When $\mu$ has density $f$ with respect to a measure $\xi$ and each $\pi(x,\ )$ has density $g(x,\ )$ with respect to a measure $\eta$, $\mu\times\pi$ has density $(x,y)\mapsto f(x)g(x,y)$ with respect to the product measure $\xi\otimes\eta$.