I know this scenario from my old mail provider. To get the mails you could use POP or IMAP, but there was no web interface. In order to change your passwort, you simply ssh'd into the mail server, and you'll get prompted to change your password. After that you are logged out:
Changing password for fooman.
Old Password:
New Password:
Bad password: too simple
New Password:
Reenter New Password:
Password changed.
Connection to mailserver closed.
Achieved:
- easy password change (without web interface)
- customer doesn't get a shell on the mail server
However, I don't know how common this setup is nowadays.