I've heard that this was a thing that happened among other unfortunate products of Caligula's madness. However, I've also heard it debunked a few times as something that was more of a practical joke and less of an actual government decision.
How true is the story that Caligula made his favorite horse one of the consuls of the Roman Empire? What became of this?
The general consensus I've found on this is that he might have and if he did it was much more likely a rational decision than a mad man appointing his horse to high political office. Most of what we "know" about Caligula comes from people who lived after him and probably had a bone to pick or just made stuff up so the accuracy of their portrayal of him is in question. That being said if he did appoint his horse it was a way of basically telling senators and other politicians that they were pointless and he was the one calling all the shots. Kind of a so easy a caveman could do it thing but in this case a horse. So if it is true it's likely just a political statement/insult and not the decisions of a raving lunatic. Unfortunately we may never know the full truth of what happened and why. Source: Caligula, A biography by Aloys Winterling.