SHA-256 "almost unique"? I have seen numerous references on the internet of people describing SHA-256 as generating an "almost unique" hash.

Exhibit A. there are more.

Is there some mathematical basis to the almost uniqueness? I would have thought that as SHA-256 is an evenly-distibuted (?) hash algorithm which can accept any message (infinitely many) as an input and produces a fixed length (finite) hash - then there will in fact be an infinite number of each possible output - which I would think hardly qualifies as "almost unique".

Is there a basis for this statement?
Theoretically, since the domain of SHA-256 contains $2^{2^{64}-1}$ different messages and the value set only contains $2^{256}$ different message digests, there must exist at least one possible output that has more than one possible pre-image.

Another important point is that SHA-256 is a deterministic function. This means that if you hash the same message twice, you will get the same digest both times. Hence, "almost unique" should be understood as meaning that SHA-256 will likely produce a different hash for each different input message. This might be false in an abstract mathematical sense, but it is probably true in a more practical sense:

In practice, uniqueness is not determined by the abstract theoretical non-existence of collisions, but by the practical non-existence of collisions. In order to find a collision in SHA-256, you would probably have to execute the algorithm some $2^{128}$ times. It is unlikely that this will happen anytime soon, even if you count the total number of times SHA-256 will ever be executed by anyone in the entire universe combined.

Does a message "exist" if it belongs to a well defined abstract set, or does it "exist" because it has actually been produced and has been represented by someone or something in the physical reality?