I think, the comment of lucas is the best answer; I'll elaborate a bit
If I mix these two containers eventually both the Oxygen and Nitrogen will be at the same temperature. Why is that?
By temperature you mean the mean energy per particle. We observe experimentally, that if you mix two heaps of particles with different mean energies, after some time the energies will have the same mean. Even if the particles are completely different (see disclaimer below), i.e. have very different masses as oxygen and hydrogen.
I mean: the mean taken over the particles of one species will be equal to the mean taken over the other species. I'd say, that this is not at all trivial (correct me if you see something), if you consider them in the mechanical model as balls that collide all the time.
I'd like to stress: only this fact allows us to define temperature as the mean energy. This would not be a meaningful quantity, if the energies of a mixture would not equate!
Of course it is more correct to say, that the energy per degree of freedom will be at equillibrium. In the case of $O_2$ and $N_2$ this coinsides with an equal energy per particle.