This has nothing to do with the birthday paradox. That paradox presumes a small countable state space (365) and a large enough # of observations.
In this case, it's a mathematical fact that 2 random vector in high dimensional space is very likely to be near orthogonal.
A slightly stronger (and more relevant) statement is that the number of mutually nearly orthogonal vectors you can simultaneously pack into an N dimensional space is exponential in N. Here “mutually nearly orthogonal” can be formally defined as: choose some threshold epsilon>0 - the set S of unit vectors is nearly mutually orthogonal if the maximum of the pairwise dot products of between all members if S is less than epsilon. The statement of the exponential growth of the size of this set with N is (amazingly) independent of the value of epsilon (although the rate of growth does obviously depend on that value).
This is pretty unintuitive for us 3D beings.