Earlier quoted context omitted.
UID per se no. Asking for it, most definitely yes. This is a hill I would die on. I have a passport and driving licence. I can be asked to produce my driving licence in limited circumstances (suspected of driving offences and stopped) at a police station within 7 days. Every other production of ID is optional, except passport at international ports. No ID needed for NHS care, dental care - there's an NHS number but I…
Note that, right or wrong, this is the kind of attitude that made Brexit happen.
Ireland doesn't require ID either. You could visit Dublin on the ferry from England showing no papers other than your ferry ticket, bought for cash as you walked on to the boat, and I'm pretty sure vice versa. Even when there were the armed checkpoints between North and South. Did this quite a few times. I see no significant support for an "Iexit", so it can't be connected with ID, or attitudes to it, but the disingenuous politicians and right of centre British media stories on the EU. Murdoch et al.
As far as I know no EU citizens were required to have additional identity requirements, or present anything for NHS treatment, until the actions of the current and previous Tory governments. Which I found offensive in the extreme. FWIW I voted remain.
If I travel to a country in the EU or elsewhere that requires ID, I carry ID. I'd have my passport anyway. I might wish they had a similar attitude to ID, for the reasons stated.