The government has all the information, but the UK has never (despite a few attempts, by Labour every time) had a national ID card and therefore no one in authority has ever had the right to demand that an individual 'shows their papers' or proves how they are, unless they want to interact with the government, or they commit, or are suspected of committing, a crime
In simple terms, the British approach has always been that public servants are just that, they are there to serve the public, not exert control over them. The suspicion, justified based on an increasingly authoritarian government, is that if carrying an ID card, digital or physical, ever becomes mandatory then its is not inconceivable that the government gives all sorts of minor functionaries in local government, as well as the police, the right to demand that one identifies oneself, and makes not having one a crime, at any time.
For many that's an over reach - and given the predilection for your local authorities and police to get carried away - be that spying on parents, fining people tipping coffee down a drain, monitoring social media it's a perspective that is understandable.
Now, you might say that its fine in other European countries, one of which I live in, but there are a few reasons for that. Firstly, they are not on the whole islands, with the mentality that brings, secondly they have almost all been invaded in modern times and so ID cards were a necessity, and thirdly the organisations that can demand you produce ID are clearly defined and regulated.
One could argue that there's no need for an ID card to achieve what they claim to want to achieve, since everyone in the UK with a right to work should have an NI number But, the government has lost control of that data set, with many NI numbers issued to people who aren't entitled to one or don't have the right to work in the UK, and those belonging to people who have died or lost the right to work not being cancelled. In most organisations the solution would be to fix the data set and process you currently have, not simply create a new one which, presumably, has the same risk of going wrong. But, tag the word 'Digital' on it, and suddenly it becomes more robust ;)
Also, calling it a Brit Card sounds a little too National Front for many people, like the misjudged naming of your railways to Brit Rail. Riffing off the '90s New Labour popularity and brit pop presumably...underlining how out of touch they are 😂