By clicking twice on an animal before the combat cooldown for shooting appears, you can get two shots off, with the second shot ignoring cooldown.
I discovered this on a taming attempt with a horse. I misclicked shooting it twice, and it ruined the taming entirely, since the first shot ended the taming process, and the second one re-aggroed it. This was done on Ender client, so I moved to default client to do some testing: https://streamable.com/puylyx
In the video, I'm able to shoot the badger twice, even while fleeing, by clicking on him while the first shot's projectile is still in the air. I repeat the process twice with a boar as well. I think this is because the cooldown doesn't apply immediately, but rather only when the creature is damaged. Since you are able to queue another shot in that small window of the first shot firing and hitting, you can queue a second shot, and, therefore, get two shots off at once.
I did not test to see if this works with arrows, but I would suspect so as long as you can fit in the click in between the mentioned timeframe.