This is a question we're seeing from our long term customers that were use to seeing them separated. There is a short answer and a long answer.
The main purpose for combining them is to clean up our inventory SKUs and make signing up for a caliber easier and simpler. Just because the SKUs are combined doesn't mean you'll always get either caliber in every situation. So while 223 and 5.56 NATO (or 308 and 7.62 NATO) are not "technically" the same rounds we treat them as similar enough for inventory purposes. We DO match the purpose to the round however - so if you select "Practice" you can get either option but if you select "Hunting" you will only get the civilian counterpart (223 or 308 respectively). On the other hand, if you select "Tracer" (when available) then you would get the military version - because is is all that is available. "Competition" (ie: Match) is the only spot where there are usually both options available and you might get the civilian version or military version depending on what is available at the time.
First on the topic of the ammo we ship the most: FMJ "Practice" rounds. If you have a semi-auto rifle chambered in either 223 or 556 (as in the barrel is stamped "223 Remington" or "5.56 NATO") you can use either round in the gun. (if you have heard otherwise, here is a link with pressure testing results).
The same goes for nearly every 308 / 7.62 semi-auto (but there are some nuances here). Even manufactures like PMC and Aguila realize this and will interchange between "308" and "7.62" in their ammo with the exact same dimensions and bullet weights. So for all intents and purposes, with standard weight FMJ: 147-150 grs for 308/7.62 NATO and 55-77 for 223/5.56 then there is no difference. (in our world this categorized as "Practice").
When SHOULD you be concerned about differences? As far as the ammo is concerned: Hunting rounds and Tracers.
For 308 / 7.62 NATO you could see problems if you are trying to shove a big 308 hunting round into a surplus rifle that is marked 7.62 NATO - it wasn't designed for that. The hunting bullet will be longer and probably higher pressure than a typical FMJ and jam into the rifling to create excess pressure. (the same problem doesn't exist for a 5.56 rifle running 223 because the 556 chamber was designed for longer bullets (ie: tracers).
In practical terms, for our situation while we have combined 223 / 556 and 308 / 7.62 NATO, if you have selected "Hunting" as your purpose you will only get your civilian counterpart (so 223 or 308 respectively). There is no common 5.56 "Hunting" round available - everything will be .223 - the same goes for 308, there is no 7.62 NATO hunting round. So even though you have a 308 chambered bolt action you can subscribe to our SKU: RF-308762-HUNT-MED (308 Win / 7.62 NATO Winchester Medium Game Hunting) and be confident that you are only getting 308 ammunition. We fit the specific caliber to the purpose.
The same is true for military 64 grain Tracer rounds in 5.56 which are longer than a typical bullet and could be an issue in a 223 marked rifle with a shorter freebore - creating excess pressure. You should only shoot these in a 5.56 NATO marked rifle.
There are many good articles out there on the topic and their technical differences, and you can spend countless hours on forums debating the issue.
Ultimately it is up to you, the gun owner to know what your gun can and can't shoot. Our service is designed for the majority of gun owners to make it easy to "set it and forget it" and get ammo that fills a particular caliber/purpose. For those that feel they want finer control for certain caliber/purposes, my advice is: don't subscribe to those through AmmoSquared and save them for direct purchase at the store or online.