Horses and dogs have pedigrees that can be traced to prove their purity even if they don't look right for their breed; chickens are not pedigreed in any way because it is impractical to try to do so with the way chickens are kept. Appearance is therefore everything in the chicken world. If it doesn't look like a specific breed, technically it is not that breed, even if both parents were. Most people will still consider it that breed, referring to it as mixed color or off-standard, but by the rules of the standard, it is not technically that breed because its appearance is incorrect. That is by the
strictest definition and I am not necessarily saying I agree with it.
Something worth noting is that traditional Easter-eggers from hatcheries aren't even mixed-color Ameraucanas. The Ameraucana breed was selectively bred from what we know of today as Easter-eggers back in the mid 1900s, with the founding breeders selecting for a specific set of traits to make a uniform breed. Hatcheries cling to the name because that's just what they were called before the Ameraucana breed was standardized in specific colorations, just like they were all called Araucanas before that breed was standardized. Since it is now in the standard and those birds do not conform to the standard, they are, as a technicality, not Ameraucanas. It has been long enough that hatcheries should have adapted, but some stubbornly refuse to and continue to cause confusion with the breed.
Another thing I think is worth noting is that only the very strictest will say that if the variety is not in the standard, it's not an Ameraucana. A lot of people recognize project varieties as Ameraucanas, as long as they are an actual variety with the proper shank and skin color, proper comb, are bearded, etc. and not just a mixed variety or some other sort of mutt that someone is trying to pass off as a true breeding variety. One of the 'rules' for a variety that is acceptable to the standard is that it must breed true 50% of the time. In other words, breeding that variety together should consistently produce at least 50% of that variety again. So with Blue as an example, Blue x Blue gives you 50% Blue and 50% not Blue, but it is considered to breed true.
A lot of people are really strict about it, though, because the Ameraucana is a
relatively new breed (admitted in the late 1970s as I recall, but I don't know the exact year off the top of my head), and with some hatcheries still selling their Easter-eggers as Ameraucanas despite how long they've been in the standard, and naive keepers buying them and also trying to breed and sell them as Ameraucanas, the breed struggles a bit. I think it's perfectly understandable as someone who is passionate about a breed that people get confused about a lot that you'd be very protective of that breed and what makes a specific bird 'count' as one. Being that I'm working with silkied Cochins, which are frequently accused of just being Silkie x Cochin mixes when they are most assuredly not, I fully understand this struggle.
Anyway, sorry, I've been lurking and thought I would try to explain a few things as I understand them regarding why things are the way they are with this breed in particular.