Transport providers (ferry companies and airlines) may require ID. As has been noted, some do so even within the UK. Not having a driving licence, I have to show my passport when flying from Wales to Scotland, for instance. I don't have to do so if I travel the same route by train.
So what about if you travel between, say, Wales and Ireland on your own craft? (I know leisure sailors who do this.)
This is what the Government website www.gov.uk/government/publications/notice-8-sailing-your-pleasure-craft-to-and-from-the-uk/notice-8-sailing-your-pleasure-craft-to-and-from-the-uk says about leaving and arriving on UK shores:
"Are there any immigration requirements when I arrive in the UK?
Anyone on board who is not an EU national must get a UKBA officer’s permission to enter the UK from a place other than the Isle of Man, or the Channel Islands. As the person responsible, you must make sure that anyone requiring immigration clearance (including yourself if appropriate) obtains the necessary permission to enter.
If there is anyone on board requiring immigration clearance, they will need to contact the nearest UKBA office by phone to arrange clearance. The National Yachtline will be able to advise on how to do this."
and
"Are there any immigration requirements when I depart from the UK?
You do not need to notify an Immigration officer of any voyage where the first port of call overseas is expected to be in the Channel Islands, the Irish Republic or elsewhere in the EU."
I read this as meaning that if everyone on board is an EU national, no immigration notifications are necessary when arriving, and that none are needed when sailing for a destination in the EU. Note that this is true despite the UK and the ROI not being in the Schengen area.
What will happen once the UK leaves the EU? I will try to get my sailing acquaintances to nag their MPs about this.