.223 Remington has SAAMI spex established 1962. SAAMI = Sporting Arms and Ammunition Manufacturers Institute (the European equivalent is CIP Commission Internationale Permanente pour l'Epreuve des Armes à Feu Portatives ). CIP has standards for .223 Remington are similar to SAAMI but slightly different too. I suspect .223 Remington sporting spex are tailored for bolt action rifles. (I could be wrong and someone will tell me I am.)
Military contracts determine the specification of 5.56x45mm ammunition which includes the pressure level at the gas port for semi or full automatic firearms. Each different contract.
My understanding is that the main difference between .223 and 5.56mm arms and ammo is the chamber dimensions (particularly the throat or leade where the bulllet enters the rifling) and the rate of twist in the barrel.
I have heard most .223 barrels are usually throated for bullets in the 36 to 55 grain range, with twist rate of 1 turn in 12 inches, whereas most 5.56mm NATO is throated for bullets in the 55 to 62 grain range, with typical twist rate 1-in-9 although the first M16 and AR15 rifles were 1-in-12 for the 55 grain.
Firing the heavier bullets in chambers with shorter leade can cause pressure to go up. Same precaution as loading highpower rifle cartridges with bullets that touch or enter the rifling. Watch the primers for excess pressure signs: if they are observed, stop. True for any caliber, any gun, any lot of ammo.
I also have heard that for countries that forbid military calibers, firearms are often marked .223 Remington rather than 5.56 NATO with no real difference.
PLUS there are some AR or Mini14 barrels marked .223 that are truly .223 Remington, but many are actually identical to 5.56mm NATO.
It's like using +P in a revolver: you won't know for sure unless you ask the manufacturer if the gun or barrel was made for it.
I have treated .223 and 5.56 ammo factory loaded with 55 grain FMJ as equivalent. Some will tell me not to worry, others will advise me to make sure I have good insurance.