"By taking the temperature readings in degrees C, you, in fact, are taking the "base data"."
I don't see how? It is a matter of how you define "the base of the data". My point is that there is a calculation on the measurement, and only the result of the calculation is recorded. Hence the base data is discarded.
We don't know what calculations were done on the climate date in question here. But it seems at least possible that the calculation is also trivial (for climate researchers anyway, could be some standard procedure), just as to us converting heights to degrees is trivial.
Recording the type of thermometer is another matter - likewise the climate researchers will probably record where their measurements were taken, which implies the equipment used.
Edit: of course if the calculations are reversible is another issue. I see now that is the point you made with the thermometer (calculate height from make and degrees). Well, it was just an example, and we don't even know yet if the calculations in question here are not reversible. I suppose if they calculated an average they are not. But I also think there are probably many cases where a measurement is just taking an average (probably even the height of quicksilver is an average in a way...).