The purpose of a password is to have an authenticator that is human-readable/writeable and, ideally, human-memorable. Ideally, a strong password is stored only in a person's head; in the very worst (and unfortunately common) case, it's also stored in a password manager.

API authentication doesn't have the memorability problem, because the password has to be stored somewhere the client program can reach it. But, as you can see, it does have the storage problem, which means you need to account for the fact that it might be compromised in storage.

So you need a separate credential for API access. Since it's only going to be used by computers, there's no purpose served by making it anything other than a computer-usable secure credential; hence: a 128 bit token from /dev/urandom.

Once you have a 128 bit token, there's also little purpose served by forcing clients to relay usernames, since any 128 bit random token will uniquely identify an account anyways.