Ron,
The K-36 tenders consist of two major units, namely, the tank/coal bunker and the frame/running gear. All of the original tank/coal bunker units have already or soon will reach the end of their useful service lives. I don't know if the original Baldwin tank assemblies had any identifying numbers on them, but I do know that the frames do.
Next time you get to Chama or Durango, look carefully on the right side frame rail of the tender, just aft of the locomotive cab access ladder. There you will see stamped into the web of the frame rail a number such as 1234 1/4 E68. Baldwin used a numbering system to identify the locomotive/tenders that they built. Each of the digits describes attributes of the locomotive, such as number of wheels in pony truck, number of drivers, number of wheels in trailer truck, and I believe, bore and stroke. No doubt someone out there can shed more light on the Baldwin numbering system for you. Anyway, what you are interested in is the last two digits following the E. For the K-36 class, 68 was the first of the series, the 480, while 77 was the last of the series, the 489.
If you are in doubt about which locomotive number corresponds to the Baldwin number, look on the back of the valve crosshead guide casting and you'll see the Baldwin number stamped there. The number is also stamped on the boiler steam dome near the opening but is normally covered by the steam dome jacket.
I would guess that the musical tenders game started about...oh, say 1926 when the first annuals started coming due. Back in the 1970's, I believe only one of the nine remaining locomotives was connected to its original tender frame, and that no doubt was by pure chance.
Sincerely,
Mark Yeamans