It doesn't need a small number -- rather it relies on you being able to find a pairing amongst any of your candidates, rather than find a pairing for a specific birthday.
That's the paradoxical part: the number of potential pairings for a very small number of people is much higher than one might think, and so for 365 options (in the birthday example) you can get even chances with far fewer than 365, and even far fewer than ½x365 people..
I think you're misunderstanding. If you have an extremely large number like 2^256 you will almost certainly never find two people with the same birthday (this is why a SHA256 collision has never been found). That's what the top-level comment was comparing this to.
We're not using precise numbers here, but a large number of dimensions leads a very large number of options. 365 is only about 19^2, but 2^100 is astronomically larger than 10^9