[Yahwe]

To be honest it's not a question of why is it admissable. It's a question of why it might be inadmissable.

Since the institution of the Wolfe Reforms and the Civil Procedure Rules (which came in while I was still an undergrad so about 99/2000) there are almost no instances of inadmissable evidence in civil claims (what you think of as 'suing').

The situation is different in a Criminal court where admissability is a big issue.

The only civil one I can think of off the top of my head would be if you sued the government and they said the evidence was 'classified for national security' but even that is a question of evidence being un-obtainable not inadmissable.

I've had a flick through The White Book (the official publication of the CPR) and I can't see anything to prevent it's admissability.

One of the founding tennants of the CPR was to clarify civil evidence rules and Lord Wolfe did this by saying "all evidence is admissable" on the basis that there is no jury and judges are professional enough to deal with everything.

Recorded evidence is certainly admissable. I've used it in court myself.

So then comes the question of why your insurance company says no. I can think of the following answers:
1) You're not talking to a lawyer.
2) The person you are talking to can not be arsed to make the necessary copies of the recording (especially from a phone)
3) they're thinking about evidence rules pre-CPR
4) They have misunderstood the Data-Protection and Freedom of Information Acts.

EDIT: Now the judge might hear the recording and still find against you. Defence counsel would certainly argue that he was under stress/it was a instinctual reaction/blah blah blah.

But it is certainly admissable.