Apart from custom "match" rifles and carbines, there are three common chamber variations for AR barrels:
1. 5.56MM NATO - Permits firing of 5.56MM NATO and .223 Remington (with slightly reduced accuracy).
2. .223 Remington - Permits firing of .223 Remington (5.56MM NATO can produce chamber pressure spikes depending on how close the chamber size is to specification).
3. .223 Wylde - Permits firing 5.56MM NATO and .223 Remington with roughly equivalent accuracy and without a potentially dangerous pressure spike associated with firing 5.56MM NATO.
It makes no difference what markings are on the receiver of a firearm. Marking a receiver "multi-caliber" doesn't make the barrel any more capable of withstanding the firing of a particular cartridge. Pay special attention to the marking on the barrel. If your barrel isn't marked, contact the manufacturer to be sure your rifle or carbine can fire the type of cartridge you intend to fire.
For example, a Colt AR6721 has a barrel marked "5.56 NATO / 1/9 HBAR". The receiver of the same firearm is marked "Cal. .223". The firearm is chambered for 5.56MM NATO ammunition and will also fire .223 Remington.
Also, while some cartridges may chamber in a 5.56MM NATO barrel, that doesn't mean a particular firearm is suited for firing them. The extra heavy 5.56MM long range bullets (heavier than 77 grains) come to mind. A bolt gun may feed such cartridges, but your AR magazines may have problems with them.
Generally speaking, 5.56MM NATO ammunition from 55 to 77 grains will work just fine in a 5.56MM NATO chambered barrel with an appropriate twist rate.
Generally speaking, .223 Remington will also work just fine in a 5.56MM NATO chambered barrel, with slightly reduced accuracy.
Generally speaking, firing a higher pressure, dimensionally variant cartridge like the 5.56MM NATO in a weapon chambered for .223 Remington is not a particularly good idea.