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I always assumed it was a bit like SHA hashing. Yes, collisions are theoretically possible. But they're so unlikely that it can be used as a unique identifier for most purposes.
That is not at all what this article is about. The headline is terrible.
The research is suggesting that there may exist "per-person" fingerprint markers, whereas right now we only use "per-finger" markers. It's suggesting that they could look at two different fingers, (left index and right pinky, for example) and say "these two fingerprints are from the same person".
When they say "not unique", they mean "there appear to be markers common to all fingerprints of the same person"
The truth is more interesting than the headline