This expression is used ironically to indicate an event, often an instance of human tragedy, or the subject of that event, was treated as merely a number in discussions of that event (i.e. tragedy) type in the public forum.
Strictly speaking the usage is incorrect: a "statistic" is a function on data. The complaint is not that the person or event has been treated as a statistic, but as a data point.
According to the Bayesian foundations of statistics, we can legitimately apply the standard methods of statistics (e.g. fitting the distribution of data to a normal curve) if we can treat data points as exchangeable. This follows from de Finetti's Theorem.
If we can't treat data points as exchangeable, however, if we know more relevant features of one data point than another, then in principle standard techniques and Bayesian techniques (which take into account prior knowledge) can produce radically different analyses.
So, when someone tells a tragic anecdote and bemoans that it is "just another statistic", they tell us more about that data point, one which had perhaps been the object of "statistics" which delivered an incorrect analysis without the benefit of the prior knowledge provided by the particulars of its case.