If you need a shorter hash just truncate a modern hash algorithm down to 160 or 128 bits. Obviously the standard lengths were chosen for a reason, but SHA2-256/160 or SHA2-256/128 are better hash functions than SHA1 or MD5, respectively. Blake2b/160 is even faster than SHA1!
(I suspect this would be a good compromise for git, since so much tooling assumes a 160 bit hash, and yet we don't want to continue using SHA1)
Just as a note, the primary reason for the truncated variants is not to get a shorter hash but to prevent extension attacks. For variants without truncation, the final hash is the entire internal state, therefore an attacker can calculated the hash of any message that starts with the original message and then has additional content without knowing the original message. Truncating the hash denies access to the complete internal state and makes this impossible.
Another way to prevent extension attacks is to make the internal state different whenever the current block is the last block, as done for instance in BLAKE3 (which has as an additional input on each block a set of flags, and one of the flags says "this is the last block").
Git has already implemented a solution based on SHA-2 with 256 bit output so that's unlikely to be changed for the time being. (But it has not really been launched in earnest, only as a preview feature.)
(I suspect this would be a good compromise for git, since so much tooling assumes a 160 bit hash, and yet we don't want to continue using SHA1)