No; in-universe, the term was used generically in the 22nd century according to Enterprise and in the mid-23rd according to the 2009 film...So the use of "Starship Class" as some sort of unique designator was always awkward. It's well-forgotten.
We also know, thanks to ENT, that by the time the NCC-1764 Defiant was built, the plaque now says "Constitution Class" and not "Starship Class." So at some point between the construction of the Enterprise and the loss of the Defiant in "The Tholian Web," the nomenclature was changed (at least for the Defiant).
Also, the Xhosa's plaque reads "Antares Class Cruiser." Most likely then, it was built after the TOS Enterprise, since its plaque might have just read "Cruiser Class" to match with the Enterprise's 'Starship Class," instead of describing the actual class name of the ship as would be done later.