It's a matter of probability distribution
It's not that every particle has energy proportional to the temperature, but statistically, particles are more likely to have speeds around the average temperature. Indeed, velocities are distributed according to Boltzmann distribution.
A liquid at Thermal Equilibrium
By studying the speed of thermally interacting molecules in a liquid, and by using the Boltzmann distribution, someone who has done a bit of Statistical Physics can derive another distribution called Maxwell-Boltzmann distribution (see image below from Wikipedia) which holds for thermally interacting particles which can exchange energy. Consider a pool of water at thermal equilibrium. Maybe, you expect to find few particles with kinetic energies $T<<K_bT$, few particles with energies $T>>K_b T$ and many with kinetic energy around the value $K_b T$. And this is indeed the case (image below)
Image taken from Wikipedia
A system not at Thermal Equilibrium
A situation of systems not at equilibrium for instance is a very weird pool where half the particles have zero speed and half the particles have a lot of kinetic energy. The particles are not interacting and hence their speeds don't change as a result of collisions with each other. Then the probability distribution does not follow the "special one" (Maxwell-Boltzmann distribution) and the system cannot be said to be at equilibrium.
To Recap:
In the case you described of the two systems at thermal equilibrium it means that both physical systems have molecules with speeds distributed according to the same probability distribution. This probability distribution is parametrised by the temperature. Hence if the systems have the same temperature, they have the same probability distribution and are at thermal equilibrium.
It is not entirely true that there is not heat exchange if the two systems are put in contact: indeed there might be small fluctuations, however on average the net heat exchange will be:
$$\langle Q\rangle_{time} = 0$$
