That’s easy to do. You just check that the username exists. If someone enters a wrong username/password pair, you can still check that the username exists, but how do you know that the user intended to log in with that username? You would also have to check every other username to see if the password matches, and that can’t be done with a simple search because you need to compute a different hash for each user you check. Then if the username exists and the password also happens to match someone else’s password, then what do you report? Should you even report it? Because doing so reveals that someone had that specific password, and if the list of usernames is publicly available (which they often are, or could become public through a leak of some sort), then you can brute force over a small set of usernames to match them up.
I keep hearing that, yet the websites will gladly tell you that the username is taken when trying to register
There are also a lot of websites where you first just enter a username and only when that is valid they ask for a password
That’s easy to do. You just check that the username exists. If someone enters a wrong username/password pair, you can still check that the username exists, but how do you know that the user intended to log in with that username? You would also have to check every other username to see if the password matches, and that can’t be done with a simple search because you need to compute a different hash for each user you check. Then if the username exists and the password also happens to match someone else’s password, then what do you report? Should you even report it? Because doing so reveals that someone had that specific password, and if the list of usernames is publicly available (which they often are, or could become public through a leak of some sort), then you can brute force over a small set of usernames to match them up.