With people so lawsuit happy that you get end up in court for serving "too hot" coffee, the ambulance chasers would be salivating at the opportunity to sue the gun industry if there were any real world safety issues with 5.56 vs. .223
That is precisely why they put the warning.
Just like many gun manuals say not to reload even when they are little different from other guns.
We are in a time period when warnings are no longer realistic and instead cover even the unrealistic and unlikely to reduce liablity.
This unfortunately can also have the opposite result in causing people to not know which warnings to heed and which are simply lawyer speak to reduce liability if something did go wrong. Which in turn has made many product instructions and labels worthless.
It should never be an issue.
Even the slightly different potential pressures are minimal when considering thier percentage difference of the overall pressure, and compare it to the proof loads both are expected to safely fire.
The primary potential issue is the lead. If you put a really heavy long 5.56 in a .223 and the bullet is or is nearly touching the rifling prior to being fired it will be operating beyond intended pressure. You could potentially have an issue with a 70+ grain round in a .223 jammed against the rifling.
That same rifle though will likely have a twist rate that cannot stablize such a heavy bullet anyways, and so choosing that ammunition would result in poor results.
Many companies also give a lead closer to 5.56 even in thier .223 labeled rifles. This may be for compatibility or to reduce liability. The downside is it can reduce accuracy with lighter weight bullets, reducing performance for those getting a .223 for actual varmint hunting. The longer jump to the rifling results in less predictable results, and an inch difference between shots when hunting a rodent could be significant.
or .308 vs 7.62 for that matter
.308 and 7.62 are the reverse, with the civilian round being the higher pressure one.