So you are right that the random number becomes "password-like", but of course if you really care about that you don't need to store the exact random number, just as you don't have to store a raw password.
However, if your random login code is easily typable then it's usually drawn from a small enough keyspace that any such code is trivially brute-forceable. Like if it's decimal digits you need more than ten.
So in practice people just trust that you are on good terms with your email provider and anyone else with access to your email, and use other mechanisms to limit access to these codes from insiders.