Does currency in one C172 model apply to other models?
@RyanGriffith is correct on the legal standpoint however its worth elaborating on your other point a bit 


  This could be the case for a flying club that has many different
  versions of the same aircraft model. Perhaps on a nice day of flying
  the aircraft that you have been flying more recently has already been
  booked. You would still like to perform the flight with a different
  aircraft also offered by the club. However, it has been more than 3
  months since you have last flown that specific aircraft.


Most flying clubs, FBO's and anywhere else you may rent an aircraft from can impose other currency requirements and in many cases their insurance may require it. So while you may be legal to fly an aircraft that does not mean you always can, and in some cases if you do it may not mean the insurance will cover it should something go wrong. For example the flight school I fly out of has both fuel injected and carbureted 172's if you are checked out in the fuel injected one you are allowed to fly the carb'ed one but not the other way around. They also have similar requirements for their Cherokee's where you can fly anything with less HP than what you have been checked out in, for example I am checked out in their archer so I can fly that as well as their 161 warrior. 

So in the specific case you mention for a flight club you should further confirm what their operating rules are.