To be clear when I say "private unique identifier" I mean an identifier, which is not stored on the database for the website, it would be hashed in combination with the password and a good salt in the hash table for login information to each account. And when a new user signs up or has forgotten their unique identifier a new one would be emailed to them, and said email along with all trace of the identifier would be deleted from the server after hashing it with the password and storing them. To clarify further you couldn't use a display name because then it would be connected to the account on the hash table, and you couldn't use an email address because the server needs to keep that unencrypted in case you lose your account credentials and you need to be sent a reset link. This would be something like the initials of your name with a unique identifying alpha-numeric or numeric code tacked on to the end which is too long to be brute-forced.
Now the reason why I think this should be an obvious idea for account security. If you were a malicious actor who managed to steal the entire hash table with salts and related account information from a website then there are certain attacks on the said hash table you could use to gather login information. This includes hashing the most common passwords list in combination with the salt values to find the easiest logins. This attack obviously wouldn't work so well on accounts with safe passwords, but I'm going to look even further for maximum security even for accounts with bad passwords. If you have a private unique identifier hashed with the passwords, then dictionary attacks using common password lists wouldn't work because they don't have the unique identifier, and it's not user-chosen so there would be no way to find it.
I've gone over this method a few times in my head and I feel like it's pretty solid, and it has probably already been named somewhere, but I'm not sure what it would be called and I couldn't find it. One thing I know for sure is none of my accounts use a method like this. Please let me know what the name of this practice would be, what attacks would work on a security method like this, and why more websites don't use a method like this. The only reason I can think of is that asking users to keep track of a unique identifier in addition to their passwords all for a security method that only adds so much to passwords is more security than most websites care to use.
Thanks for any and all answers.