The output length/size of a hash function is fixed, whereas it takes an arbitrary-length/size input.
That doesn’t necessarily mean you can figure out what length output for a given input is needed to make it one to one. Not sure you could avoid collisions even if the length of the output was infinite, but I’m assuming different inputs have to have outputs that diverge at some point.
Assume you have two inputs, A and B.
A may hash to: A38uT75kjGz B may hash to: A38uHso629t
So yes, if you were to cut these off after the A38u then you would no longer be able to say for sure if you hashed A or B to arrive at your hash.
Of course in practice this usually isn't a problem as long as you have "a long enough" output.
Assuming a perfectly random uniform distribution, the usual desirable property of a cryptographic hash - the probability of a hash function not having a fixed point (that is, hashing at least one x to itself) is (1-1/n)**n, where n is the number of possible outputs. As n approaches infinity - which it does pretty rapidly in this case, since we're talking about 2**32 to 2**512 in practice - this approaches 1/e, or about 37%.
So, not only is it possible, but most "good" hash functions (63% of them) will have them.
We use different words for Cryptographic Hashes, Password Hashes, XOFs, MACs, and (non-cryptographic) Hashes because they do different things and have different security properties. Misusing the terminology makes reasoning about what is meant difficult.
Suggested reading: Sponge Functions - https://keccak.team/files/SpongeFunctions.pdf
"Informally speaking, a random oracle* maps a variable-length input message to an infinite output string. It is completely random, i.e., the produced bits are uniformly and independently distributed. The only constraint is that identical input messages produce identical outputs. A hash function produces only a fixed number of output bits, say, n bits. So, a hash function should behave as a random oracle whose output is truncated to n bits."