> all information that exists could be stored in a storage medium big enough

Why is quantization necessary for information storage? If you're speculating about a storage device external to our universe, it need not be constrained by any of our physical laws and their consequences, such as by being made up of finitely many atoms or whatever. It might have components like arbitrary precision real number registers.

And if you're speculating about a storage device that lives within our universe, you have a contradiction because it's maximum information capacity can't exceed the information content of its own description.