i.e. that there should be no need to also carry your passport, unless intending to travel on outwith the EU?
However, when I'm traveling to somewhere that has a hotel safe, or somewhere secure enough to leave valuables, I'll bring the full passport booklet in my carry-on along with me, and leave it there when I go out. That way, in the unlikely event I should lose my wallet / passport card, I still have the full booklet to travel home on, rather than having to arrange emergency travel documents for the flight back.
I haven't needed it yet, but I don't see the harm in also taking it. It's not as if it takes up an obscene amount of space in my carry-on luggage, nor am I risking losing it by carrying it on my person at all times. It's purely there for peace of mind, and a backup in the very unlikely event I need it.
The above is only true in the Schengen zone. As soon as you cross the border to a non-Schengen country or you arrive from a non-Schengen country you will go through normal border control (with some exceptions). This includes multiple EU countries beside Ireland like Romania, Bulgaria or Cyprus. And many non-EU countries like Bosnia, Serbia, Monaco, Kosovo, Andorra, Ukraine, Turkey, Russia, etc.
(I'm exaggerating, but only slightly.. the irony is that I used to have data roaming included in Switzerland, but the network forced me to a new plan that doesn't have that because the old one was almost but not quite complying with EU roaming rules)
Nothing in EU is as simple, therefore it's not only a Schengen/non-Schengen thing. For example a border between North and "regular" Ireland isn't marked in any way, you just cross a brook or a field.
For anybody unfamiliar with Europe, I just want to add that there aren't any checkpoints or border crossings, you just drive through or walk over. Half the time you won't even notice you're doing it. You're just supposed to have your ID on your person when you do.
Ireland (and previously the UK) is the only major exception to this, as it's part of the common travel area alongside the UK. In practice, this doesn't make a huge difference when you're an Irish or EU citizen (it does if you're a tourist who needs a different visa to visit), as it's an island, and airlines require a form of ID to board anyway. When travelling from Ireland to Schengen, I usually have a passport in my carry on luggage as a backup, and take my passport-card (similar to a national ID card) in my wallet.