I wrote about Caligula's horse here a while ago.

That Caligula made his horse consul or wanted to make his horse consul is something that pops up from time to time, that as best I can tell, is the result of bad reading. Suetonius is the source for the incident, and clearly says it's a rumor, and the context Suetonius sets that comment in makes me confident it's a joke at Caligula's expense.

It's popular to try to rehabilitate Caligula in part by making this horse business a joke Caligula pulled on the senate - a measured insult. Winterling in his biography of Caligula uses it this way, and is a good example of the pro-Caligula use of this story:

Dio reports how Caligula invited Incitatus (“Hotspur”), his favorite race horse, to dinner, fed him barley corn made of gold, toasted him with golden goblets, and planned to make him a consul. [Dio is copying Suetonius here]....

There is no way now to know whether everyone in Rome got this joke. Nor can we tell whether Suetonius understood it later or—as seems more likely—failed to understand it on purpose, since he makes use of it to present the emperor as insane.

As I note in my linked comment, this is a bad reading because Suetonius says it was a rumor, and Winterling here presents the affair as a fact. Suetonius is careful to distinguish his facts from his rumors, and I have to think Winterling either overlooked Suetonius outright stating this was a rumor or chose to credit the story despite Suetonius saying it's a rumor.

It's a popular story and it's possible there's an argument for the pro-Caligula position as a good reading, but I don't know it.