The feature is called "Login Verification", I think, and it's only SMS based, no Google Authenticator / Authy style one-time password... Also, it was saying I needed to verify my email address before that feature can be used, but there was no option to verify the email address that is used since I've registered almost a decade ago... Had to change my email (used the username+somestring@gmail.com trick to reuse my address, as one email can be used with only one twitter account...) then change it back.