The chamber throat thing comes from back when most 223 uses shorter bullets under 55 grain, where the M193 556 uses longer 62 grain bullet. If there is not enough chamber throat for the longer bullet, pressure will go up drastically.
From what I understand the slight pressure difference between 556 and 223 is resulted from different measuring method between SAAMI vs CIP standard.
In term of the brass, they are identical in external measurements, so there is no 223 or 5.56 brass. The internal volume of each brand of brass may vary. In general, lower internal volume results in higher pressure with everything else being equal (same powder charge). This is not something you need to worry about unless you reload.
I have seen American Eagle 223 and 556 have very different point of impact from the same AR, which usually indicates there is some pressure difference. But again this was experience from one gun, one brand of ammo.
My take is generally speaking the 223REM barrel is unlikely to blow up if a few rounds of 556 is put through it. But you will find out if your rifle does not like the particular ammo (hard bolt lift, action not cycling, poped primer, heavy extractor mark on case head etc). If that's the case, stop shooting that specific ammo from that specific rifle.
The reality is 223 or 556 can mean different things to different ammo maker, and same with barrel maker. You will not know unless you try them.
In practice, I often wonder why people even bother with 5.56, as it usually costs much more than 223.