If the similarity of ciphertexts reveals information about the similarity of the cleartexts, then given you have the encrypted password and the encryption algorithm, couldn't you guess the cleartext password by performing search? The point about hashes is that nobody can decrypt, although everybody knows the "encryption" algorithm. So not sure that homomorphic encryption would help in this case. Or am I missing something?

Comparing the hash values would also work (if done right, hash collisions are extremely rare).