The cool thing about this is that it's self-balancing - if either side gets larger than the other due to random chance, the ball in that side will have more space to bounce in, and therefore bounce less often, slowing its growth. Meanwhile, the ball in the smaller side will bounce more often in its smaller space, making up the ground.
There are stableish equilibria that are not 50-50, e.g. one color having a donut around the other color that has a donut hole.
Now the question remains, are there stableish equilibria that are 50/50? Splitting it into two half-circles sounds like an equilibrium at first glance, but I'm not convinced it is, as only a tiny bit of random luck seems to make it become a "horseshoe" pattern instead.
(That assumes that the simulation is randomized of course, which doesn't seem to be the case for the one in the link posted here.)