It doesn't matter at all if one person's hashed password is identical across two of that person's accounts on two different websites. The identical hash will instantly let an attacker (with access to both hashes) know that this person shares the same password across two accounts. But that is of no value; the attacker is going to start by assuming that it's true anyway.

Salts are there to ensure that two accounts on the same website which have identical passwords nevertheless have different password hashes.