All the EU countries except for Denmark, Ireland and UK have identity cards.

All the EU countries except for Ireland and UK are part of the Schengen Area (or are obliged to join in the future, for some recent members). It remains to be seen what Ireland status is once UK is out, depending on what kind of border is put in place between them.

The trend in the EU is for more control (regarding IDs and in many other issues) and this was not well appreciated by many UK citizens.

That neatly demonstrates the idiocy of Brexit: The UK, Ireland, Channel Islands and Isle of Man have had a common travel area since before WW2, requiring no papers. Which could easily disintegrate with the way our govt is going about Brexit, and would cause issues. Complete freedom from papers changed 15 or 20 years ago for air travel in that it got silly: you "had to prove that you didn't need to prove your ID, by having Irish or British ID". The ferry kept on requiring nothing at all, I don't know if that's changed since.

The trend in the EU has been for removing ID to travel between countries within the zone. As far as I am aware once you're inside the Schengen zone ID rules remain whatever is the case within the country. i.e. Denmark wasn't required to make ID compulsory to be within the system, it's only supposed to be travel in and out of the zone that's affected is it not? There don't seem to be moves to change. Obviously a Dane would carry some ID when visiting France or another EU country requiring it.

Course there's all the "temporary" internal borders inside Schengen for migrant or terrorist checks etc that weakens the whole concept.

My point is that the UK is a bit of an outlier in the EU and was resisting many of the “unification” policies (including also the common currency, again in the company of Denmark — and Sweden that’s supposed to join but it’s just not doing it). UK citizens distrust their government and even more a supranational government and in the same way that UK citizens don’t like being controlled by their government the UK doesn’t like being controlled by the EU. That’s what I meant by “this kind if attitude” and apparently these are hills a majority of people chose to die on.

Ireland - yes and no.

The Irish Government like to deny it, but I'd argue that the "Public Services Card" (https://psc.gov.ie/) is a de-facto ID card (and hence central ID database).