differential equations - Does every ODE comes from something in physics?

A fairly silly answer is that the answer is obviously "Yes", since one can build a computer to integrate numerical solutions to your ODE. (now that I think about it, Sigfpe's answer is essentially the same as mine.)

Going along these lines I guess one can find more "physical" models of my suggestion (in the sense that doing physics is often finding toy models that contain the essence of the phenomena etc) by proposing various lattice models or cellular automata which are known to have universal constructors. Or by designing circuits made out of balls and springs.

I spent a little bit of time trying to put down the right words which would make your question more precise and more in line with your intent, but I think ultimately it boils down to what kind of physical models you'd be satisfied with.

As much of physics can be described in terms of ODEs, any sufficiently powerful type of model is going to contain the sort of answer I described above. I think the right question is what's the "simplest" (or perhaps "weakest") known physical model for Painlevé VI.

One kind of answer to that question would be finding a physical system for which some solution of Painlevé VI gives some physically measurable function - along these lines, I know that Painlevé functions are highly useful in various integrable models / lattice models, e.g. famously, the spin-spin correlation function in the 2D Ising model in the scaling limit is a solution to Painlevé III - thus, my guess is that Painlevé VI shows up in one of these contexts, but the literature is pretty vast.