The USS Constitution NCC-1700 was the first Constitution Class. Unless there was a USS Constitution before the commissioning of the Constitution Class that I am unaware of.
First ship of the class tends to be named the same as the class.
USS Intrepid, Intrepid-Class
USS Galaxy, Galaxy-Class
USS Defiant, Defiant-Class
etc.
Of course, this does not rule out the possibility of other ships being called Intrepid, Galaxy or Defiant. In fact, in the three examples I stated above, at least 2 ships have been seen in TOS but as different classes then what they are most known for. USS Defiant, I believe was a Constitution Class or a variant of her.
In the case of the USS Constellation, they screwed up bigtime giving it a lower registry then that of the USS Constitution. Throughout the Trek timeline, it has been common practice that the first ship of the class be called the same as the class name. Thus, the belief by some people that the USS Enterprise refit from The Motion Picture, was a Enterprise-Class. Most likely other ships in this era also were refitted and they kept the classname of Constitution Class.