Largest Australian ISP using private Twitter account information for identity
twitter.com
twitter.com
In fact they are probably doing so specifically because people were already social engineering them this way.
Sure the stuff they are asking for is ridiculous, but that's because you are attempting to use non-standard channels of communication.
You're getting mad at them for trying to meet you halfway.
Pick up the phone. Send them an email. Use their contact form and have them contact you.
This is all easily avoided.
Personally I would never allow anything related to anyone's account to happen over Twitter DMs unless there was something like a system in place where you give them your Twitter handle during sign-up.
Like others have experienced they're sending unsolicited SMS spam that cannot be opted out from with a simple reply.
Regardless of all this - the information they're asking for is not acceptable, especially since they're a telecommunications company that could simply SMS, call or send a push notification to the Telstra app that is installed on your phone.
Lastly, please don't make assumptions that everyone can just "pickup a phone a call them".
Why does every brand need to seem like a teenager on social media? It's an ISP, people buying their product don't need to "identify with their brand".