Given a database where we have usernames and passwords, we want to secure users' passwords by hashing them. We should not use only username and passwords in this hash, as someone having data from other sites that uses the same hashing function can come to the same results and thus know our passwords. If we try hashing the passwords by themselves it would be easy to bruteforce the entire database and guess all the passwords. In normal applications, passwords are hashed with a random salt that is stored along with them to prevent them from being bruteforced easily.
However, we could just use username, password and the same salt for every record in the database. This option would save one data field for each record, while the combination of username and password would be unique for our entire database. Is such a solution just as safe as the traditional password hashing scheme, or are there some security drawbacks to them I am not aware of?