The birthday paradox relies on there being a small number of possible birthdays (365-366).
There are not a small number of dimensions being used in the LLM.
The GP argument makes sense to me.
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..
You expect to find a collision in 2^256 possibilities in ~sqrt(2^256) = ~2^128 tries.
You expect to find a collision in 10^10000 possibilities in ~sqrt(10^10000) = ~10^5000 tries.