A passport is not an identity document, it is a permit to travel.
A driving licence is not an identity document, it is a permit to drive.
Not everyone has either of them, and it's unfair to expect people to carry them about unless they are travelling abroad or driving. Especially if they are ill.
We do have NHS numbers in this country, which is a hangover from WW2 identity legislation, but we do not use them enough. I think it would be absolutely fine to be given a credit-card-sized thing with your NHS ID number on to show at medical appointments, or an adapted EHIC card which has the EU provision listed on it as well, and I can't imagine any British person thinking that was a big deal, even normally anti-ID types. If you have forgotten it, you should be asked to phone it through afterwards unless it's patently obviously a repeat appointment and that you clearly live here and have done so for over a year.
If medical personnel have reason to believe people would be at risk or that there would be a major problem trying to get hold of their NHS card/number, for example because they are away from home and have forgotten it, or in an abusive relationship, or are in A and E separated from their documents, staff should be able to tick a box and ignore the fact it's not there, but perhaps flag it for future reference, to be checked next time.
And the Nigerian woman with quins? The NHS has an annual budget of £116,400,000,000 which means she cost 0.000001% of that. Not exactly going to make a big difference to the nation's health, that shortfall.