No, that's not correct at all.
That line in "The Doomsday Machine" only means that all ships with that configuration (whatever configuration that Spock had in mind, by the way) are starships. It does not mean necessarily that all starships have that configuration.
If I see a Mustang and say, "By configuration, a sports car," it does not mean that all sports cars are Mustangs.
This is the dialog:
PALMER: Sir, I'm picking up a ship's disaster beacon.
KIRK: Try to raise it, Lieutenant.
SPOCK: I have it on the sensors, Captain. By configuration, a starship stopped in space. She appears to be drifting.
http://www.chakoteya.net/StarTrek/35.htm
So the configuration that the
Enterprise and
Constellation shared was only used by starships. I guess it is possible but not proven that some starships had different configurations. Spock did not identify the class of starship. So either there was only one class of starship or Spock was not yet able to see the configuration clearly enough to tell what class of starship it was.
Spock's words seem to prove that other Federation civilian and starfleet interstellar ships that were not classed as starships had a different configuration than starships and Spock was already able to tell that the
Constellation had a starship configuration and not the configuration of another type of ship.
In "Bead and Circuses":
CLAUDIUS: You're a clever liar, Captain Kirk. Merikus was a spaceship captain. I've observed him thoroughly. Your species has no such strength.
MERIK: He commands not just a spaceship, Proconsul, but a starship. A very special vessel and crew. I tried for such a command.
http://www.chakoteya.net/StarTrek/43.htm
Clearly a starship is a much greater ship than a civilian space ship, and probably much greater than the average Starfleet ship. Starships are probably
Constitution (and maybe other) class ships (and not necessarily all the members of those classes) that are armed equal to (or perhaps are members of) the top classes of warships, and also equipped with research facilities equal to the best pure research ships, and are assigned to special types of missions where both qualities may be needed, such as five year missions, for example.
Thus we need not assume that the Federation has only 12 or 13 starships total at a time for exploration, or that its defense force has only 12 or 13 top level warships.
And thus there is no need to assume that all of the starships seen in TOS belong to the
Constitution class since it is possible that other starship classes closely resemble the
Constitution class on the outside.