I've never heard the term hatching eggs used to refer to chicks that have just hatched. Everyone I've seen advertising eggs or chicks for sale uses the term "day old chicks" to denote sending chicks in that 48 hour time period after hatch before they need food and water. The term "hatching eggs" is used to denote eggs that are assumed fertile (and usually are, although some breeding set ups or pairs of birds for whatever reason do not produce 100% fertile eggs so there's always a chance that a few of them may not be fertile) and sent to you for the purpose of you putting in your incubator to hatch.
As for "hatching eggs" being a grammatical error when used to describe the kind of eggs, technically it is not. This is one of many instances where the rules of English are bent and an action word is used as an adjective to describe the intended purpose of the noun instead of a verb. Like sewing needle, cooking wine, marking pencil, walking stick, or gardening gloves. In many languages, grammatical rules state that in order to denote the purpose of a noun the noun must be followed by the purpose, such as "gloves used for gardening." But in English it is common and acceptable to use the -ing involving term as an adjective in front of the noun. Therefor, "hatching eggs" becomes a correct structure because the -ing involving term "hatching" is a verb being used as an adjective to describe the eggs. This inconsistency in grammatical use of terms is one thing that makes English somewhat challenging for some non-native speakers to learn, but it is correct use of the English language.
