You shouldn't need proof of identity to switch to a different account with the same provider. But, if you open an account with a new provider, there are strict money laundering rules designed to prevent the recycling of money from drugs crime, terrorism, and so on.

The bank or building society must check you are who you say you are and that you live where you say you do, so they will ask you to provide separate proofs of identity and address. Official documents are preferred. You may need more proof for phone or internet accounts.