Hi Odea,
I made a Storm claim earlier this year and assessor needed bank details to finalise and pay. This was external assessor for Zurich. I think it is relatively standard practice - he just wanted to issue complete sign-off.
Also noticed that recently banks and building societies are nearly always asking for ID when making withdrawals at a cash desk so it should be safe giving account details. As mentioned, you already give people your details if you write a cheque....