[zahllos]

You can sort of look up a birth certificate but the service isn't designed for that. It is here: https://www.gro.gov.uk/gro/content/

This is where you get certified copies should you ever need that for interfacing with foreign governments that want them (the European country I live in very much wants a copy of my birth certificate).

It's not an identity check by any means but a legitimate birth certificate ought to be findable here.

But yes birth cert + utility bill is a very, very weak binding to identity.