...Which may simply be the same as "AEGIS class": a description of what the ship does or carries, separate from any proper-name identity. The Enterprise
class would simply be Starfleet's first and only starship built for the NX mission, with NX gear, hence the rare opportunity to use that other unique identifier.
Indeed, when Archer first uses the expression "NX class" in "Fortunate Son", he uses it to describe the power of his starship, just like a USN skipper would brag on having an AEGIS class vessel. The Boomers or the Nausicaans wouldn't know Enterprise
class from Lollipop
class, but they may have heard of the NX project...
NX class is UESF's only known starship design without a proper name - as opposed to Neptune
class and possibly Triton
class. The later UFP Starfleet only ever operated one starship class that didn't appear to have a proper name, and that may be because the training ship was a civilian design otherwise known as the J class. Or then there indeed existed a USS Jay