IMHO in a modern navy one might say that all battleships are in the battleship class, or type, or sort, or kind, or set and all aircraft carriers are members of the aircraft carrier class, or type, or sort, or kind, or set, and all destroyers are in the destroyer class, or type, or sort, or kind, or set, and so on.
That is speaking very generally. Almost all battleships of the US Navy have hull numbers in the series from BB 1 (USS Indiana 1895) to BB 64 (USS Wisconsin 1944).
US Navy aircraft carries have hull numbers from CV-1 (USS Langley, 1922) to CV-67 (USS John F. Kennedy 1968) or CVN-68 (USS Nimitz 1975) to CVN-77 (USS George H.W. Bush 2009).
But a class of vehicles is usually used in the sense of a set of vehicles built to the same set of plans and specifications with only comparatively minor modifications.
For example there are hundreds of cruise ships operating today, but only three operating ships of the largest class, the Oasis class, and three of the next largest class, the Freedom class.
At the present time all the aircraft carriers in service in the US Navy are Nimitz class, but three of the Ford class are in stages of construction.
During World War II the US Navy had four Iowa class battleships, two North Carolina class battleships, four South Dakota class battleships, three Colorado class battleships, two Tennessee class battleships, three New Mexico class battleships, two Pennsylvania class battleships, and so on.
IMHO in the era of TOS any civilian or Starfleet spaceship that traveled between stars was a starship in the usual sense of the word, but in the eyes of Starfleet members the word starship was reserved for large powerful ships made for exploring and protecting the far frontiers of the Federation. Not just spaceships or starships but STARSHIPS.
If the Constitution class starships were the first starships to be built they might have been listed as Starship class Starfleet vessels until the next class of starships was introducted, and then they might have been renamed Constitution class starships without changing the plaques on the bridges.
Or possibly the plaque on the bridge states that the Enterprise is a member of the Starship class, or type, or sort, or kind, or set, of Starfleet vehicles without giving the further detail that it is a member of the Constitution class subset of the Starship class - even though according to 20th or 21st century practices it would be usual to describe it as a Constitution class instead of a Starship class.