AFAIK the contrary is invalid, so that "not every output is the result of one and only one input".
Generally when people make this claim, what they're actually referring to is what's called Collision Resistance (CR) and/or Weak Collision Resistance (WCR), which instead make claims on difficulty of finding such collisions (of which infinitely many exist).
WCR, necessary for almost any cryptographic use, states that for any given input it should be difficult to find a different input which hashes to the same value. CR, generally desirable for cryptographic hash functions, states that it should be difficult to find two different inputs such that their hashes are equal. CR implies WCR, but WCR does not imply CR -- for example, SHA-256 (currently) exhibits CR but SHA-1 only exhibits WCR.
"So every piece of data in the world has its own unique hash digest."
This is false for the reasons apeescape describes: every piece of data in the world has its own hash digest, but these hash digests are not unique.
I'm saying that for a layman explanation, it's reasonable to say that hash outputs are unique. Because following that with "technically, it's more 'practically' unique, theoretically there are collisions but you won't encounter them with probability > 2^-256" (or whatever it is) just confuses the topic to them more than just summarizing. You have to admit that most people won't go on a 200h adventure to learn about the state space of 256+ bits and how to conceptualize tiny statistical probabilities, so there must be a point where you have to cut the explanation to an approximation of the truth. This is true in every field.
The air molecules in the room you're in can't all gather in one corner because that's not possible, it's forbidden by conservation rules.
But they won't gather in two opposite corners only because that's so tremendously unlikely, it would be allowed by conservation but statistically it's ludicrous.
The same is true at the opposite end of the spectrum. Almost all real numbers are normal (in all bases) but the nature of "Almost all" in mathematics is different in an important way from "All" and I want people to grasp this difference when I'm discussing properties of numbers. It definitely is not true that all real numbers are normal, you probably rarely think about any normal numbers at all.
I agree. I think this wording would be better than in my previous comment, what do you think?
it's reasonable to say that hash outputs are *almost surely* uniqueYou could have said exactly the same thing about MD5 right up until you couldn't. Then you could have said "oh yeah well MD5 is broken, but it's safe to assume you'll never find one for SHA-1", right up until we did. So if you say "oh yeah well SHA-1 is broken, but it's safe to assume you'll never find one for SHA-256", I disagree.
It would be one thing if collisions in hash functions were found by just repeatedly hashing things until you find a collision. If that were the case, then yes, I'd agree with you on those 1-in-2^256 odds, at least for a while. But by and large, that's not what happens. Over time, weaknesses are found in algorithms which allow you shrink the search space, which significantly changes your odds.