You don't need game mechanics
You can just do it! If you agree that your PC can be made up of three owls in a trench coat, you can just say that it's true. You're the DM, after all.
For example, I had a player who wanted to be a persimmon tree, for some reason. We eventually decided that she would use the mechanics of a Wood Elf, but would be a living persimmon tree for all other purposes. Doing so opened up a lot of fun RP potential, while not being gamebreaking.
As for a gameplay justification, you don't really need one. D&D is full of magic that the players cannot access. For example, sentient magic items exist, but the players have no way to create one. If you peek into published adventures, you can see examples of mysterious, plot-driving magic that is inaccessible to the players (e.g. Out of the Abyss). You shouldn't overuse magic as a plot device, but they're fine for situations like these.
If you really want game mechanics...
There are a few ways that this could be done using the existing game mechanics, or variations on them. First, the Wish spell and direct divine intervention can accomplish pretty much anything. Also, a slightly modified True Polymorph is capable of permanently transforming a creature into another one, though it can't change them into multiple creatures by RAW. The Deck of Many Things has a Void card, which draws out your soul and puts it into anything of the DM's choice. Your character could have pulled this card and been split into 3 owls. As you can see, though, there aren't many RAW options that let you create such a PC.