Very nice question! ;) I wrote a short paper about this question about ten years ago, see http://arxiv.org/abs/math/0206017 (My apologies for advertising my own work, but this is exactly the question I asked myself at that time).
The product of probability spaces is tensor product in the sense of category, as Martin Brandeburg also pointed out in a comment. But it has an additional structure, you have natural morphisms onto the factors in the tensor product. This is because the projections onto the two factors that you have for the Cartesian product (of sets) preserve the measures, so they are also morphisms in the category of probability spaces. I called this structure a tensor product with projections: for two objects $\Omega_i=(\Omega_i,\mathcal{F}_i,P_i)$, $i=1,2$, you get $\Omega_1\otimes\Omega_2=(\Omega_1\times\Omega_2,\mathcal{F}_1\otimes\mathcal{F}_2,P_i\otimes \Omega_1\otimes\Omega_2=(\Omega_1\times\Omega_2,\mathcal{F}_1\otimes\mathcal{F}_2,P_1\otimes P_2)$ and random variables $X_i:\Omega_1\otimes\Omega_2\to \Omega_i$, $i=1,2$.
You can use this "tensor product with projections" to characterise independence of random variables: two r.v. $Y_i:\Omega\to\Omega_i$, $i=1,2$, defined on the same probability space $\Omega$, are independent iff they factorise, i.e., if there exists a r.v. $Z:\Omega\to\Omega_1\otimes \Omega_2$ such that $Y_i=X_i\circ Z$, $i=1,2$.
The notion dualises to the algebras of functions on a probablity space, where it becomes a tensor product with inclusions. Generalising to not necessarily commutative algebras, it includes notions of independence used in noncommutative (or quantum) probability, like the freeness.
