You shouldn't need proof of identity to switch to a different
account with the same provider. But, if you open a savings 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.