I have a whole pile of FC brass only stamped "FC 12" (FC on top, 12 on bottom) that was packaged in the black American Eagle boxes that are labelled .223 Rem. The headstamps don't have a "223" anywhere on them.
A friend of mine who bought an ammo can of Federal 55gr "5.56x45mm NATO" has identical headstamps on his. Technically it shouldn't say NATO since it's 55gr I guess? Federal/American Eagle sells a lot of 55gr "5.56x45mm NATO" ammo.
I don't know if the two identical headstamps packaged in different boxes chrony the same or not.
My LGS was selling cardboard boxes of "5.56" ammo a year or two ago from Federal. I know several people who bought some and they said it chronied a tad over 3000fps and shot fireballs out of the muzzles of several different rifles. All had the same headstamps; "FC", year number, no NATO cross, no "223 REM".
So it would seem from my experience that Federal either uses the same generic FC headstamps for both .223 Rem and 5.56NATO ammo or sells the same ammo as both ".223 Rem" and "5.56x45mm NATO". Either way, without a box, there's no way to tell.