I don't get it either. The author is adding a common dependency to two independent series makes them correlated, which to me seems trivially true. He then goes on to say that this form of correlation is uninteresting, because we're only interested in whether variations are correlated, which doesn't seem so trivially true: after all, a linear trend is a first-order variation, and if two things both increase at the same rate, then their relationship is worth at least a second look.
It looks like what the author is really trying to say is that we should pass the data through a high-pass filter, eliminating any 'expected' trends such as inflation, and instead observing if the noise of the two datasets is correlated. This is an observation that has some value, but is certainly not trivial to pick a threshold for the high-pass (certainly it's not always just a linear trend), and the mutually dependent variable can have as much noise (if not more) as the two measured data sources, so you still might get "false" correlation.
It looks like what the author is really trying to say is that we should pass the data through a high-pass filter, eliminating any 'expected' trends such as inflation, and instead observing if the noise of the two datasets is correlated. This is an observation that has some value, but is certainly not trivial to pick a threshold for the high-pass (certainly it's not always just a linear trend), and the mutually dependent variable can have as much noise (if not more) as the two measured data sources, so you still might get "false" correlation.