I believe that in the Catholic Church in Ireland the title of archdeacon is, as you say, honorary, and is simply a title conferred upon a senior member of a cathedral chapter.
In the Church of England it is also fairly common for an archdeacon to be a residentiary canon of the diocese’s cathedral, but in that case the role of archdeacon is still very much substantive and will be the archdeacon’s main job.
One unusual title in the Church of England is that of archdeacon of Westminster, which is largely an honour conferred upon one of the canons of Westminster Abbey. The position does come with some real responsibilities, but these are not really comparable with the duties of an archdeacon responsible for an actual archdeaconry.
The Royal Navy, Army, and Royal Air Force, as well as HM Prisons, also have archdeacons, a position held by the most senior Anglican chaplain, who may or may not also be the most senior chaplain.
Some dioceses also appoint archdeacons for specific functions rather than with jurisdiction over an actual archdeaconry.
Finally, there are also honorary archdeacons in Anglican dioceses around the world. In practice, the title of honorary archdeacon often seems to be conferred upon a British clergyman by a diocese in a less developed part of the world as a gesture of gratitude for some kind of charitable contributions. The main advantage of being an honorary archdeacon is that it confers the style “the Venerable” and the right to be addressed as “Mr Archdeacon”, which seems to matter to some people. It does lead to the somewhat peculiar situation where an ordinary parish priest is going around calling himself “the Venerable” and asking to be addressed as “Mr Archdeacon”, much to the confusion of those who do not know that this is because he has been made honorary archdeacon of Mbandaka or something like that.
