I'll correct the math tomorrow. Thanks for the heads up. The principle remains the same though.

Well, there are only so many hairs in the inner ear to transmit sound.

https://en.wikipedia.org/wiki/Hair_cell

"The human cochlea contains on the order of 3,500 inner hair cells and 12,000 outer hair cells at birth" Since those hairs are necessary to transmit sound to the nerve cells - that does create a limit to number of audio imputs that can be transmitted to the brain.

I didn't find anything to clarify how many hairs must react to register a sound at all ( hairs to decibel for example). But given that the number of hairs is much higher than the maximum number of decibels the ear can process . . . So there is a limit on the number of audio inputs that can reach the neurons.

Then,from there, how much of that data can the brain process? It gets complicated, because not only does the brain recognize pitch, it process elements like rhythm and volume. Our brains are limited, so short answer is yes, there is a limit to how much of any kind of input the brain can process.

I didn't find anything that gave a number for audio input. But, I did find this very detailed explanation of how neurons connect to the hairs that transmit sound.

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3078955/

"This doesn't necessarily relate to how many different sounds you hear, "

Thanks for noticing that your reply doesn't address the question.