They can ask for what they want to investigate. He does not have to provide them if he chooses not to.
I cannot tell you what they are thinking and why. I believe they think he is one and the same as the person that took out the phone.
Your son in law should cooperate with them to have his name cleared but he should take it further to have the notation removed immediately.
It is not him and therefore it should not be on his record. Very straight forward.
This should all be reported to the police as well by your son in law. This is fraud.
If he has verified himself he should not have to do it again.
Tell him to go to the Police and also the TIO
Write to VEDA and set out what he knows and demand his name be cleared.
If he is losing money because he cannot get a loan he should work it out and seek compensation.
He should not focus on what Telstra is telling him. He has to focus on what he knows and what he has to do to make this go away.
Do not do it over the phone. Do it all in writing and keep copies.
Until Telstra see he is serious they will not do anything.
Also threaten to go to the media. This is also a story they love to show and Telstra will not be shown in a good light either.
I hope this makes more sense?