So, putting myself in the opposing army's shoes here:
The first battle is probably a rout - You've found a new weapon, no-one knows how to deal with them, the Rhino charge wins hands down.
The second battle, being a sneaky medieval commander, I show up prepared. Rhinos have soft feet, so I get my smiths making hundreds of caltrops* to scatter. I cut longer pikes, and dig indentations so the shafts can be braced into the ground. I have my bowmen carry sharpened stakes, and knock them into the ground facing forwards. I try and find boggy and marshy ground, and try and persuade you to attack me there. As I'm not the one feeding muti-tonne animals, I can probably afford to wait around there.
If we have a tradition of horse archery, I send out horse archers - the Rhinos are slower, with lower stamina. We lure them across the battlefield, and every now and then one of them goes beserk.
Without horse archers, I get my cavalry to feint - pretend to charge, and then withdraw. With any luck, the rhinos will chase them, with their riders unable to control them, and I can lure them across the battlefield until they're too tired to stand, and then deal with their riders with lances. This works even better if my cavalry have a few javelins to pelt them with.
Light siege weapons would work very well here - a ballista has very good range, and just a few crazed rhinos running back through the opponent's lines might make them question using them in future
They'd also be hit hard by the standard medieval tactic of retreating, and burning all the crops, forage etc you can find along the way.
In short, they're a terrifying weapon, but easy to counter if you base your entire strategy on them.
*A quote from Vegetius, writer of Epitoma rei militaris:
The armed chariots used in war by Antiochus and Mithridates at first terrified the Romans, but they afterwards made a jest of them. As a chariot of this sort does not always meet with plain and level ground, the least obstruction stops it. And if one of the horses be either killed or wounded, it falls into the enemy's hands. The Roman soldiers rendered them useless... they strewed the field of battle with caltrops, and the horses that drew the chariots, running full speed on them, were infallibly destroyed. A caltrop is a device composed of four spikes or points arranged so that in whatever manner it is thrown on the ground, it rests on three and presents the fourth upright.