I'm not sure what would that accomplish.
What would prevent someone else than me that they reached by accident from calling them back with the ticket number that they gave him when they assumed it was me?
IMHO everybody should have government issued keypair with public part easily checkable on government site.
When they want to confirm your identity they just ask you to sign something random with your key. "To confirm your identity sir please enter the following into your government issued identity card and read to me back what you see on the display of the card."
Before giving you that message, they might ask you security questions, etc., which you'd be able to answer (because you called them back at their listed phone number and you know it's them).
I think this id card should contain private key securely embedded and public key easily readable and that government should publish public keys. To avoid leaking of private key this ID card should be able to cryptographically sign given data without revealing private key.