Well, not all hashing algorithms but all interesting or considered useful hashing algorithms, probably.

When dealing with say countable infinite sets you can certainly create a provable unique hash for each item in that set. The hash won't be interesting or useful. E.g. a hash that indexes all the integers n with a hashing function h(n+1)... so every integer you hash will be that value plus one. But this just being pedantic and wanting to walk down the thought.

In the past the FBI used some cryptographic hash. Collisions with a secure cryptographic hash are functionally unobservant in practice (or else the hash is broken).

The use of the perceptual hash is because some people might evade the cryptographic hash by making small modifications to the image. The fact that they'd discarded the protection of cryptographic hashing just to accommodate these extra matches is unsurprising because their behavior is largely unconstrained and unbalanced by competing factors like the public's right to privacy or your security against being subject to a false accusation.