Actually, ATK (used to own Federal, Speer, CCI and a slew of other brands-now spun off into Vista Outdoors) currently has the contract to operate the US Government owned Lake City Ammo Plant (You can see this on the Orbital ATK website). All the machinery, tooling, standards and specifications, etc belongs to Uncle Sam. So, the Federal ammunition products are not produced on the same equipment as Lake City ammunition and never were. (Historical note: back in the 1960's, Olin/Winchester had the contact to operate the plant. Their use of ball powder in early 5.56mm ammo created massive problems.)

However, since a whole slew of folks make both .223 or 5.56mm, it doesn't make a lot of sense to have two sets of stamping dies for case forming, along with separate gauges and inspection standards. It'd be a liability nightmare. Folks who don't make both might show differences.