[JJM Ballantyne]

"By recognising their right to consider themselves Irish, then does that not mean that the state recuses itself from considering them British?"

Not at all. It simply means that the state (the UK) isn't concerned if they're dual citizens.*

This part of the agreement is not as significant as it sounds* and has far less to do with the UK than with the Republic of Ireland. For the UK, it's a no-brainer: if you're from NI you're automatically a British citizen anyway.

However, it's certainly not impossible that, if NI were to be incorporated into the ROI at some future date, it might ultimately lead to a loss of British citizenship.

* There are no qualifications on British citizenship that "preclude" citizenship in another country. So giving people in NI the right to consider themselves Irish actually changes nothing.