There is no law that requires passengers on domestic flights to show photo ID. Many airlines require it to prevent people transferring bookings to another person as they offer that service for a significant charge.
Security only need to check boarding passes not ID.
ID is not checked at security at any of the UK airports I travel through regularly (MAN, EDI, GLA, ABZ, BHD, SOU etc), for domestic or international flights. ID checks are done at checkin and/or the boarding gate. But technically, for a domestic flight, there is no requirement to check ID at all. I have travelled BA to London several times without being asked for ID.